Amateur tennis star Guy Haines wants to divorce his promiscuous wife Miriam so he can marry Anne Morton, the daughter of a US Senator. On a train, wealthy smooth-talking psychopath Bruno Antony recognizes Haines and reveals his idea for a murder scheme: two strangers meet and "swap murders"—Bruno suggests he kill Miriam and Guy kill Bruno's hated father. Each will murder a stranger, with no apparent motive, so neither will be suspected. Guy humors Bruno by pretending to find his idea amusing, but is so eager to get away from Bruno that he leaves behind his engraved cigarette lighter.
Guy meets with Miriam, who is pregnant by someone else, at her workplace in Metcalf, their hometown. Miriam informs Guy that she no longer wants to end their marriage. She threatens to claim that he is the father, in order to thwart any divorce attempt. That evening, Bruno follows Miriam to an amusement park and strangles her while Guy is on the train to Washington. When Guy arrives home, Bruno informs him Miriam is dead and insists that he must now honor their deal.
Guy goes to the Mortons' home, where Anne's father informs Guy that his wife has been murdered. Anne's sister Barbara says that the police will think that Guy is the murderer since he has a motive. The police question Guy, but cannot confirm his alibi: a professor Guy met on the train was so drunk that he cannot remember their encounter. The police assign an escort to watch him.
Bruno follows Guy around Washington, introduces himself to Anne, and appears at a party at Senator Morton's house. To amuse another guest, Bruno playfully demonstrates how to strangle a woman. His gaze falls upon Barbara, whose appearance resembles Miriam's. This triggers a flashback; Bruno compulsively squeezes the woman's neck, and other guests intervene to stop him from strangling her to death. Barbara tells Anne that Bruno was looking at her while strangling the other woman, and Anne realizes Barbara's resemblance to Miriam. Her suspicions aroused, Anne confronts Guy, who tells her the truth about Bruno's scheme.
Bruno sends Guy a package containing a pistol, a house key, and a map showing the location of his father's bedroom. Guy creeps into Bruno's father's room to warn him of his son's murderous intentions, but instead finds Bruno there waiting for him. Guy tries to persuade Bruno to seek psychiatric help, but Bruno threatens to punish Guy for breaking their deal.
Anne visits Bruno's home and tries explaining to his befuddled mother that her son is a murderer. Bruno mentions Guy's missing cigarette lighter to Anne and claims that Guy asked him to search the murder site for it. Guy infers that Bruno intends to plant it at the scene of the murder and incriminate him. After winning a tennis match, Guy evades the police escort and heads for the amusement park to stop Bruno.
When Bruno arrives at the amusement park, a carnival worker recognizes him from the night of the murder; he informs the police, who think he has recognized Guy. After Guy arrives, he and Bruno fight on the park's carousel. Believing that Guy is trying to escape, a police officer shoots at him, but instead kills the carousel operator, causing the carousel to spin out of control. A carnival worker crawls underneath it and applies the brakes too abruptly, causing the carousel to spin off its support, trapping the mortally injured Bruno underneath. The worker who called the police tells them that Bruno, not Guy, is the one he remembers seeing the night of the murder. As Bruno dies, his fingers unclench to reveal Guy's lighter in his hand. Realizing that Guy is not the murderer, the police ask him to come to the station to tie up loose ends.
Sometime later, another stranger on a train attempts to strike up conversation with Guy in the same way as Bruno had done. Guy and Anne coldly walk away from him.
