Michael Armstrong (Paul Newman), an American physicist and rocket scientist, is traveling to a conference in Copenhagen with his assistant and fiancÃ©e, Sarah Sherman (Julie Andrews). Armstrong receives a radiogram to pick up a book in Copenhagen; it contains a message which says, "Contact Ï in case [of emergency.]" He tells Sherman he is going to Stockholm, but she discovers he is flying to East Berlin and follows him. When they land, he is welcomed by representatives of the East German government. Sherman realizes that Armstrong has defected, and is appalled that, given the circumstances of the Cold War, if she stays with him, she will likely never see her home or family again.
Armstrong visits a contact, a "farmer" (Mort Mills), where it is revealed that his defection is in fact a ruse to gain the confidence of the East German scientific establishment, in order to learn how much their chief scientist Gustav Lindt (Ludwig Donath) and by extension, the Soviet Union, knows about anti-missile systems.
Armstrong has made preparations to return to the West via an escape network, known as Ï. However, he was followed to the farm by his official guard, Hermann Gromek (Wolfgang Kieling), an East German security officer. Gromek realizes what Ï is and that Armstrong is a double agent, and as Gromek is calling the police, a tortuous fight scene commences that ends with Gromek being killed by Armstrong and the farmer's wife (Carolyn Conwell). Gromek and his motorcycle are then buried. The taxicab driver (Peter Lorre Jr., uncredited) who drove Armstrong to the farm, however, reports on Armstrong's behavior to the police.
Visiting the physics faculty of Karl Marx University in Leipzig, Armstrong's interview with the scientists ends abruptly when he is questioned by security officials about the missing Gromek. The faculty try to interrogate Sherman about her knowledge of the American "Gamma Five" anti-missile program, but she refuses to cooperate and runs from the room, even though she has agreed to defect to East Germany. At this point, Armstrong secretly confides to her his actual motives, and asks her to go along with the ruse.
Armstrong finally goads Professor Lindt into revealing his anti-missile equations in a fit of pique over what Lindt believes are Armstrong's mathematical mistakes. When Lindt hears over the university's loudspeaker system that Armstrong and Sherman are being sought for questioning, he realizes that he has given up his secrets while learning nothing in return. Armstrong and Sherman escape from the school with the help of the university clinic physician Dr. Koska (Gisela Fischer).
The couple travel to East Berlin, pursued by the Stasi, in a decoy bus operated by the Ï network, led by Mr. Jacobi (David Opatoshu). Roadblocks, highway robbery by Soviet Army deserters, and bunching with the "real" bus result in the police becoming aware of the deception, and everyone fleeing. While looking for the FriedrichstraÃe post office, the two encounter the exiled Polish countess Kuchinska (Lila Kedrova) who leads them to the post office in hopes of being sponsored for an American visa. The police find Armstrong and Sherman at the post office, and Kuchinska throws herself in front of the police so they can escape to their next destination.
Two men approach them on the sidewalk - one is the "farmer". He gives them tickets to the ballet; the plan is to travel in the luggage of the troupe to Sweden that evening. While attending the ballet and waiting for the pick-up, they are spotted and reported to the police by the lead ballerina (Tamara Toumanova), who flew to East Berlin on the same airplane as Armstrong.
Armstrong and Sherman escape through the crowd by shouting fire. They hide in two crates of costumes, and are ferried across the Baltic Sea to Sweden on a freighter. The ballerina, desperate to reveal the fugitives' hiding place, identifies the wrong crates, which are machine-gunned while they are already dangling over the pier. Meanwhile, Armstrong and Sherman have escaped by jumping overboard and swimming to a Swedish dock.
Who escapes from school?
Armstrong and Sherman