At an anniversary party for psychiatrist Ray Flemming and his wife Carol, the doctor gets a phone call. He tells his wife that he has to go see a patient who is having an anxiety attack. His wife is upset, but he leaves anyway and goes to see another woman named Joan. Joan tells Ray she is having second thoughts about killing his wife and asks if there isn't some other way. Ray tells her that there is nothing else they can do.
When Flemming gets home, Carol tells him to sleep in the guest room and that she is calling her lawyer in the morning. Carol also alludes to the fact that she previously warned him to stop his affair or she would divorce him very publicly and create a huge scandal. Flemming insists he was with a patient and that he stopped to see another doctor on his way home. He tells Carol to call him, hoping to call her bluff, but she picks up the phone and dials. He tells her to hang up and then says that the reason he was out was because he was planning some last minute details of a surprise trip that he planned for them. She believes him.
The next day, Joan comes to Flemming's office, where they discuss the details of the trip. He gives her a dress to wear, along with a picture of his wife for her to try to match herself to. He tells her to be at his apartment no sooner than 9 PM. They discuss the details of their plan, and Joan leaves. After she goes, Flemming pulls out a pair of gray gloves
At home that night, Flemming and his wife get ready to leave on their trip. He asks her to call the maid and make sure she comes in the next day. She asks him if he's seen her blue dress, the one that he gave to Joan, but she forgets about it. While she's calling the maid, Flemming puts on his gray gloves. Carol hangs up and goes to the bar to fix a drink. Flemming walks over with his hands behind his back. Carol goes to close the doors to the balcony, and he comes up behind her and strangles her. While he is killing her, the phone rings. Burt Gordon, a prosecutor, is calling for Carol. Flemming says that she is getting dressed. He hangs up and breaks the sliding glass door from the outside. He removes his wife's jewelry, then packs up some of the valuables from the house. Joan arrives, and he tells her not to touch anything. Flemming tells her to get ready to go while he empties his wife's purse and packs a few more things. He pulls down the blankets on his wife's bed to give the appearance that someone slept there. Right before they leave, Joan calls the cleaners and tells them she's leaving a bundle of laundry outside the door. She asks them to pick it up in the morning. When she picks up the phone, Flemming puts his handkerchief over it so that she won't leave her prints. Flemming puts the laundry outside and tells Joan to add the blue dress and the matching gloves to the bundle when she gets back from the airport. As they leave, we see that the black handkerchief is still on the phone, but Flemming returns a few moments later and retrieves it.
When they get on the plane, they stage a fight. Joan gets off the plane and Flemming stays aboard. Joan watches the plane pull away, then goes to the restroom and changes her clothes. She returns to the apartment and adds the blue dress to the laundry bag.
In Mexico, Flemming goes out on a boat to fish. He asks the man on the boat with him to make some lunch, and when the man is gone, Flemming throws the things he took from his apartment overboard. When Flemming gets back home, the window is boarded up and there is a chalk outline on the floor where he left his wife's body. He is looking at it when Columbo comes out of the back of the apartment. Flemming asks Columbo what happened, and Columbo tells him that someone broke into the house and tried to kill Carol. Flemming is startled and asks if she is all right. Columbo tells him that they are only hoping she will wake from her coma long enough to talk to them. He tells Flemming he'll take him to the hospital, but Joan calls as they are leaving. Flemming pretends that it's his office and hangs up on her.
At the hospital, Flemming asks to see his wife, but he is not allowed. He knows one of the head doctors there, so he asks the nurse to page him. While they wait, Columbo tells Flemming what they know. He asks where in Mexico Flemming was. He then tells Flemming that he had to borrow a picture of Carol to send to the stewardess so that she could verify it was Carol on the plane. Flemming asks what the stewardess said, and Columbo says that she identified Carol.
Burt Gordon shows up at the hospital, where he meets Columbo. Flemming's doctor friend calls the reception desk and Flemming goes to speak to him. While he is on the phone, Gordon tells Columbo he would really like to see this wrapped up quickly. Columbo mentions to Gordon that he thought it was strange that when Flemming came home, he didn't call out to Carol. Gordon gets defensive and Columbo apologizes.
Dr. Flemming gets off the phone and says he has approval to see his wife. Columbo goes with him, but when they get to Carol's room, the officer who is watching the door won't let them in. He tells Columbo that Carol has regained consciousness, and the doctor is with her.
Flemming becomes agitated and starts pacing. When the doctor comes out, he tells Flemming that Carol passed away. Flemming asks if she said anything, and the doctor tells him that the only thing she said was his name.
In the next scene, Flemming is at the park looking at an outdoor art display. Columbo approaches him and they walk to Flemming's office together. At Flemming's office, Columbo lights a cigar and walks around looking at things. Flemming asks him if there is anything else he can do for him, and Columbo mentions Flemming's luggage. He tells Flemming that he checked the airline records and that when Flemming left, his bags were 13 pounds overweight, but when he returned, his bags were only 4 pounds overweight. Flemming tries to explain it away by saying he took along a lot of medical journals he hadn't had time to read. He read them on the trip, but he didn't bring them back with him. Columbo turns to go, but then turns back with his classic line, "One more thing..." He asks Flemming what his wife was wearing and then wonders aloud what she did with the dress and the gloves when she got back from the airport. Flemming suggests that someone stole them. Flemming's secretary calls to say that Miss Hudson is there. Columbo asks if he can come back to the apartment that night at 5:30 to check for the dress and gloves again. On his way out, he asks Joan what her name is, and she tells him.
Joan goes into Flemming's office. He starts to call the cleaners about the dress and gloves, but Joan realizes she didn't leave the gloves there, only the dress. Flemming tells Joan to go home and get the gloves and bring them to his apartment so that Columbo can find them when he returns that evening. Joan brings the gloves and tells Flemming she feels bad about what they did. Flemming tells her not to let it ruin their relationship and that they just need to wait.
Columbo shows up at the door, and Joan runs to hide in the bedroom. Columbo asks what Flemming is doing home already. Flemming says he wanted to look for the dress and gloves. He tells Columbo he found the gloves but not the dress. Columbo insists on looking again, but as he heads for the bedroom, the doorbell rings. Flemming asks Columbo to answer it because he doesn't feel like talking to anyone. Columbo answers the door and it is a man from the laundry service with the dress. Columbo goes to leave, but the phone rings. Someone is calling to tell Columbo that someone confessed to murdering Carol.
Columbo goes to interview the man, whose name is Tommy. Columbo asks a lot of questions, and Tommy gets annoyed, saying he already confessed, so there's no reason that he should have to go over it again and again. After Tommy leaves, Flemming says he doesn't believe the story. He thinks that Tommy is neurotic and only confessed because he wanted to purge himself for some guilt or to get his name in the papers. Columbo agrees with him. Flemming then suggests that Columbo staged the whole thing, and Columbo asks him "why on earth would I do a thing like that?" Flemming does not answer and goes to leave. Columbo stops him by asking if he went fishing in Mexico. Columbo mentions that where he fishes locally, people always dump things off the boat. Then he asks if Flemming took his medical journals on the boat when he went fishing. Flemming gets upset and tells Columbo that he should be concentrating on finding his wife's killer.
Flemming goes to talk to Burt Gordon, who agrees to pull some strings to get Columbo to stop bothering Flemming so much. Later, Columbo stops by Flemming's office. He tells Flemming that he's been taken off the case. Columbo then asks Flemming to take him on as a patient. He says he bothers people and makes them nervous, and he hopes that Flemming will help him figure out why. Columbo says he's too suspicious of people, and he thinks that's the problem. He tells Flemming that when he gets taken off a case, he immediately assumes someone must have applied some pressure and then he begins to wonder why. Flemming tells him to leave and threatens to go to his superiors if he doesn't leave the case alone. He then changes tone and asks Columbo to have a drink with him. Flemming tells Columbo that he thinks Columbo adapts himself to each situation, and that he pretends to be incompetent when he is actually quite intelligent. Columbo is impressed and asks Flemming to tell him what he might know about a murderer who plans every detail in advance. Flemming calls it "an elaborate, intellectual project." He then tells Columbo this murderer would be well-educated, strong, courageous, and highly detail-oriented. Columbo asks how to catch a man like that, and Flemming replies, "You don't." Then Flemming tells him that if he killed his wife, Columbo will never be able to prove it. Flemming leaves then, asking Columbo to turn out the light when he leaves.
Columbo goes to the set of a movie that Joan is appearing in. He asks to speak to her in private. He questions her about how long she has known Dr. Flemming, then asks her about her movie and whether she does her own makeup. She asks him why he is asking about Flemming, and he tells her that he thinks Flemming killed her wife. Joan says she doesn't believe it. He then pulls out a pair of sunglasses and tells Joan she looks nervous. He asks her to try on the sunglasses, but she refuses. She gets agitated and tells Columbo he has no right to tell her what to do because he's not on the case anymore. He asks her how she could know that, and she says that Flemming told her. Columbo asks how he could have told her when she hadn't seen him in a week. He continues to question her, and finally she says that she wants to call her lawyer. Columbo says he doesn't understand why she would need a lawyer if Dr. Flemming killed his wife.
He pushes her and pushes her, asking questions about the murder and telling her that Carol would still be alive if it wasn't for her. She screams at him to stop, but once he does, she regains her composure and says she has nothing to do with it. She tells him that he can't prove anything, and he says she can leave. As she goes, however, he tells her that Flemming's one mistake was her and that he will continue to hound her until she breaks down. He sends someone to follow her.
At home, Joan sees that there is someone watching the house. She gets more and more nervous and then calls Flemming. She tells Flemming that Columbo knows everything, and she needs to see him right away. Flemming tells her to wait until her appointment the next morning and to relax.
The next morning, Joan doesn't show up for her appointment. Flemming's secretary calls her house, and a man answers. He says he is with the coroner's office and that there has been an accident. Flemming tries to get more information, but when they refuse to tell him anything, Flemming rushes out of the office. He shows up at Joan's house and Columbo is there. Flemming sees that Joan is dead. Columbo says that she took a lot of pills and then drowned in the pool. He says he thinks he pushed her too far. Then Columbo tells Flemming that he's home free because Joan was the only thing that could have given him away. He says that Flemming is going to be terribly lonely, though and maybe he wants to get something off his chest. Flemming asks what he means, and Columbo says that after all his planning, Flemming still doesn't get what he was after: Joan Hudson. He did all that work so he could be with her, but now she's dead. Flemming laughs and tells Columbo he never loved Joan. He just used her. He had no plan to marry her or stay with her after the murder. As he is saying this, Joan comes out of another room. Flemming realizes the whole thing was staged and Joan, after hearing what he said about her, goes with Columbo to make a statement.