In 1919 he began studies with Alfred Borrotau, a well-respected teacher, and had his professional debut in 1920 at the Vienna Volksoper where he sang for a few years in smaller roles.
Possessing one of the largest dark-and-cavernous-type bass voices of the twentieth century, Weber was in equally high demand for villainous roles and noble characters.
In 1936 he joined the Royal Opera at Covent Garden where he sang numerous roles for several years including: Pogner, Gurnemanz, Hunding, Hagen, Daland, King Marke, Osmin, Rocco, and Commendatore.
The title role in Boris Godunov was one of Weber's favorites, and excerpts (sung in German) survive from a performance broadcast on radio.
He became a regular performer at the Bayreuth Festival from 1951 to 1962, and he also appeared a number of times at the Teatro Colón in Buenos Aires.