Wendell Armbruster Jr. [Jack Lemmon] sets off for Ischia to claim his father’s body, who died in a car accident during the island’s annual getaway to the Bay of Naples. For the past decade, Baltimore industrialist Wendell Armbruster Sr.—known to friends as Willie—has spent a month each year at the Grand Hotel Excelsior for supposed therapeutic mud baths. On the way, Wendell runs into Pamela Piggott [Juliet Mills], a traveler from London who has come to Ischia to lay to rest her mother, Catherine. Wendell learns that his father and Pamela’s mother died together in the same crash, and to his surprise he discovers that Willie had been carrying on a decade-long affair with Kate despite his Baltimore marriage.
Pamela suggests burying Willie and Kate together on Ischia, a plan that clashes with Wendell’s desire to return the remains home for a formal memorial to be broadcast to employees, the Coast Guard, and U.S. dignitaries — even Henry Kissinger — within three days. The hotel manager, Carlo Carlucci [Clive Revill], coordinates the funeral and the transportation, but a slow-moving Italian bureaucracy and laid-back traditions complicate every step. Wendell’s impatience and blunt manners irritate Pamela, the hotel staff, and the authorities.
A tangled sequence of events follows. The bodies mysteriously vanish from the morgue. Wendell suspects Pamela at first, but the truth soon surfaces: the Trotta family— Arnoldo Trotta [Franco Angrisano]— whose vineyard suffered damage in the crash, has seized the remains and demands a ransom of two million lire, revealing another “Italian tradition” of extortion. Simultaneously, Bruno [Bruno Alias], a hotel valet desperate to return to America, tries to blackmail the couple with compromising photos of Willie and Kate.
Initially, the abrasive Wendell mocks Pamela about her weight, calling her “Fat-Ass” within earshot. Yet as they reenact their parents’ old rituals around the island — staged by the hotel staff in affectionate tribute to the deceased couple — Pamela’s kindness begins to soften Wendell’s arrogance, and the two gradually fall in love. Bruno’s blackmail photos eventually include naked shots of Wendell and Pamela as they bathe together in the bay. Anna [Giselda Castrini], a pregnant chambermaid, shoots Bruno dead when she learns he intends to fled with the money and avoid marriage.
With the external pressure of the extortion scheme, Wendell’s wife back in the States reaches out through her connections to involve State Department Agent J.J. Blodgett [Edward Andrews]. Seeing the couple’s longstanding commitment, Blodgett grows sympathetic and helps push the process forward. Wendell agrees to bury Willie and Kate together in the Carlucci family vault. The problem of exporting a body is resolved by placing Bruno’s remains in a coffin labeled as Wendell Sr.’s. After a round of mud baths, Blodgett drafts Willie into an embassy post, cynically promoting “Equal Opportunity Employment” for the deceased — unaware that the coffin carries Bruno’s remains. The coffin heads home to the United States in a diplomatic pouch, where it will remain unopened during the ceremony.
Carlucci reassures them that their suite will be reserved for the same time next year, continuing the family’s tradition. Pamela vows she’ll have shed the excess weight, and Wendell quips that if she loses even one pound, their liaison will be off. The tale concludes with Wendell and Blodgett boarding a U.S. Navy helicopter bound for Rome’s airport, marking an end to one adventure and the start of another.