At age 10, Weissenberg gave his first recital, performing, among other works, an etude of his own composition. Shortly thereafter, Weissenberg and his mother attempted to flee Bulgaria for Turkey as fascist terror deepened. They were caught and thrown in a concentration camp. "Only three elements remained constant," Weissenberg recalled. "Silence, singing, and crying." What saved the pair was an accordion Weissenberg had been given as a gift by an aunt. A German guard who liked music let Weissenberg play and after three months put the Weissenbergs on a train to Istanbul, throwing the accordion into their compartment through an open window as they left.
They made their way to Turkey and then to Israel, where Weissenberg studied at the Jerusalem Academy of Music and performed with the Israel Philharmonic under the baton of Leonard Bernstein. He left his accordion with a group of children after playing an outdoor concert and departed for the U.S. in 1946. Weissenberg enrolled at the Juilliard School of Music, studying with Olga Samaroff and at times with Artur Schnabel, and making contact with Vladimir Horowitz, who urged Weissenberg to enter the Leventritt Award competition. Weissenberg won the award in 1947, and his career was launched. His U.S. debut came with the New York Philharmonic, conducted by George Szell, and for the next ten years he toured the U.S. and Europe. In 1956 Weissenberg moved to Paris, eventually becoming a French citizen. For a decade beginning around that time, he took a hiatus from performing, subjecting himself to a reconstruction of his keyboard technique. His performances of Chopin, Rachmaninov, and Prokofiev were especially notable, and the Bach Chromatic Fantasy and Fugue was a staple of his recital programming. His recordings of the 1960s and 1970s remained well represented in reissues on the EMI label as of the early 2000s, and he remained active into old age.