The midday airliner Prometheus departs Le Bourget aerodrome in Paris bound for Croydon, England, carrying eleven passengers. Among them is Hercule Poirot, the renowned Belgian private detective. Also aboard are Jane Grey, a hairdresser's assistant; Norman Gale, a young dentist sitting opposite Jane; Cicely, the Countess of Horbury, a glamorous former chorus girl with a gambling habit; the Honourable Venetia Kerr, a well-bred aristocrat; Dr. Bryant, a Harley Street ear and throat specialist; Daniel Clancy, a writer of detective fiction; James Ryder, a cement company director; and Jean Dupont and his father, distinguished French archaeologists. At the back of the car sits a stout, middle-aged woman in black: Madame Giselle.
As the plane nears Croydon, senior steward Henry Mitchell tries to wake Giselle to present her bill. Her body slumps. Dr. Bryant confirms she has been dead at least half an hour. Jean Dupont suggests a wasp sting, pointing to a wasp he killed earlier and a puncture mark on her neck. Poirot, however, notices a small thorn on the floor near the body, its tip discolored and its shaft dressed with orange and black silk. Clancy identifies it as a poisoned dart of the kind shot from a blowpipe.
At Croydon, Inspector Japp of Scotland Yard takes charge. No one saw anyone use a blowpipe. Clancy admits he once purchased one for research and is the only passenger who walked past Giselle's seat during the flight. A blowpipe is found hidden behind the cushion of Poirot's seat, bearing no fingerprints. Poirot wonders why the murderer hid the weapon rather than disposing of it through the plane's ventilator holes.
At the inquest, Giselle's lawyer, Maître Alexandre Thibault, identifies the deceased as Marie Angélique Morisot, one of Paris's most prominent moneylenders, known professionally as Madame Giselle. A government analyst testifies that the dart was tipped with venom from the boomslang, a deadly South African tree snake. The jury nearly returns a verdict of murder against Poirot but settles on death by poison with insufficient evidence to name the killer.
Poirot consults with Japp, Fournier of the French Sûreté (France's national criminal investigation division), and Thibault. Fournier reveals that Giselle combined blackmail with moneylending, gathering compromising secrets about clients and using them as loan security. This practice opens a wide field of murder motives. Thibault discloses that Giselle's fortune, over one hundred thousand pounds, was left entirely to a daughter named Anne Morisot. The investigation is hampered because Giselle's maid, Elise Grandier, burned all of her mistress's business papers following standing instructions.
The investigators evaluate each passenger. Jane and Norman are virtually eliminated by their seat positions. Lady Horbury has a strong motive given her gambling debts but could not easily have acted unseen. Dr. Bryant has possible access to snake venom. Clancy owns a blowpipe and walked past Giselle's seat. A detailed inventory of passengers' belongings yields key items: Norman's attaché case contains a white linen coat, dental mirrors, and cotton wool rolls, plus an empty matchbox in his pocket. Lady Horbury's dressing case holds cocaine. The Duponts carry hollow Kurdish pipe stems. Poirot announces that the list points to one person but calls the evidence the right clue on the wrong person, declining to explain further.
In Paris, Poirot and Fournier interview Elise, who produces a notebook with coded entries about Giselle's clients, several of which appear to match passengers on the Prometheus. Fournier learns that a blowpipe was sold to an American three days before the murder, and Poirot discovers that an airline clerk was bribed to place Giselle on the noon flight by a man in theatrical disguise who booked under a false name and never appeared.
Meanwhile, Jane's notoriety boosts her salon career while Norman's dental practice deteriorates as patients cancel out of fear. Poirot recruits them as allies and devises a plan: Norman, lightly disguised, approaches Lady Horbury as a blackmailer demanding payment to suppress evidence of her affair with a young actor named Raymond Barraclough. After Norman's visit, Poirot arrives claiming to represent Barraclough. Terrified, Lady Horbury confesses she borrowed from Giselle, who discovered the affair and threatened exposure. Cicely visited Giselle the night before the murder and begged for mercy but was refused. She lied at the inquest about never having seen the woman.
Poirot names three guiding clues: the suspiciously convenient wasp, a telling item in the passenger's baggage, and an extra coffee spoon in Giselle's saucer noticed by a steward. He concludes that the dart was not shot from a blowpipe but inserted by hand, meaning the killer physically approached Giselle's table.
Giselle's daughter, Anne Morisot, now Mrs. Richards, arrives in Paris to claim her inheritance. Poirot finds her face familiar but cannot place it. When Jane snags a fingernail at lunch, the gesture jolts his memory: on the plane, Lady Horbury summoned her maid Madeleine to bring a dressing case for a broken nail. Anne Morisot is Madeleine, who walked the length of the rear car past her own mother's seat. Anne had both motive and opportunity, but Poirot learns from Lady Horbury that the maid was not supposed to fly that day; the decision was last-minute. Before Poirot can reach her, Anne is found dead on the boat train to Boulogne, a bottle of prussic acid, a lethal poison, in her hand. The police believe it is suicide.
Back in London, Poirot assembles Japp, Norman, and Clancy for the final revelation. The blowpipe was a decoy. The real method was simpler: Someone donned a white linen coat, padded his cheeks with cotton wool, and walked through the cabin looking like a steward. That someone was Norman Gale, whose real name is James Richards, the nephew of his dental partner John Gale, whose surname he adopted. Norman met Anne at Nice while she worked as Lady Horbury's maid. Learning that Giselle was Anne's wealthy mother, he devised the entire scheme. He bribed the airline clerk, disguised himself as an American to buy the blowpipe, and during the flight donned the steward's coat, seized a coffee spoon from the pantry, jabbed the poisoned thorn into Giselle's neck, released the wasp from the matchbox as a decoy, and returned to the toilet to change back, all in minutes. He had worked on a South African snake farm under the name Richards, giving him access to boomslang venom.
Norman married Anne in Rotterdam so they could make mutual wills. He planned to let her claim the inheritance and then kill her. But when Poirot arrived in Paris with Jane, either of whom might recognize Anne as Madeleine, Norman panicked, rushed Anne out of the city, and poisoned her on the train, staging the death as suicide. Confronted, Norman inadvertently reveals that he wore gloves when handling the poison bottle, effectively confessing. Japp arrests him as James Richards, alias Norman Gale, on a charge of wilful murder. The Canadian police also seek him in connection with another suspicious death.
A month later, Jane visits Poirot, pale and shaken. He steers her toward a fresh start, having arranged for her to join the Duponts' archaeological expedition to Persia. Poirot predicts that Lord Horbury, Lady Horbury's husband, and Venetia Kerr will marry, a match he considers himself to have facilitated, and that in time Jean Dupont and Jane will find happiness together.