Weber was born in Vilnius, Lithuania, in 1913 and moved to Detroit with his family as a child. He studied clarinet with a member of the Detroit Symphony; he later studied with Simeon Bellison, the New York Philharmonic's principal clarinetist and a key figure in the development of modern clarinet performance in the United States.
Weber was renowned for his unusually rich and pure tone. Jon Manasse, principal clarinetist of the American Ballet Theater, told the Times that "it had a unique bell-like quality, that kind of clarity. The resonance of the sound, when it was correct, was enough to communicate the music without adding special effects of gimmicks."
In the late 1930s Weber was hired by Arturo Toscanini to play in the NBC Symphony Orchestra. He was principal clarinet of the New York City Ballet Orchestra from 1964 until 1986; he also had stints with the New York Philharmonic and the Metropolitan Opera Orchestra. After retiring in 1986 he taught at the Juilliard School.
His students are members of prominent orchestras such as the Cleveland Orchestra and the Dallas Symphony.
He died at Beth Israel Medical Center in Manhattan.