Weissmann was born in Vienna, Austria on August 7, 1930, to Adolf and Greta Weissmann. He immigrated to the United States in 1938, and became a naturalized American citizen in 1943. After the Bronx High School of Science, he received a B.A. from Columbia University in 1950 and his M.D. from New York University in 1954. He also pursued an early career in art, exhibiting at a major New York gallery.