Weber originally considered a career as a painter, but his splendid voice caused him to rethink that direction and concentrate on a career in music instead. In 1919, he began studies with Alfred Boruttau in Vienna and the following year he made his debut at the Vienna Volksoper as Fiorello in Rossini's Barbiere di Siviglia. In the 1920s, he sang at both Düsseldorf and Cologne before being engaged for the Munich Wagner Festival in 1931. His impressive voice and authoritative interpretations led to his engagement as a member of the Bayerische Staatsoper in 1933 and Weber spent nearly two decades in Munich singing many of the largest roles in the bass repertory. He sang the role of Holsteiner, Commandant of the Enemy Army in the 1938 premiere of Strauss' Friedenstag at Munich. From 1945 until his retirement, Weber was a member of the Vienna Staatsoper. Beginning in 1936, Weber established an association with opera in London. His Pogner in the opening night Meistersinger (April 27) was found somewhat severe, but his noble voice and superb acting were found highly commendable. Two nights later, Weber's Gurnemanz drew from critic Ernest Newman the opinion that his performance was unsurpassable. In other Wagner operas, Weber was complimented for his powerful Hunding and for his "superbly Mephistophelean Hagen." The following year, Weber returned to add Daland and to repeat his King Marke in Tristan, singled out for its originality and depth. For the 1938 London season, Weber added his richly detailed Rocco in Fidelio and his slyly acted Osmin in Die Entführung aus dem Serail, singing the role with "consummate ease."
The Second World War made Weber unavailable to London for the duration, but he made a welcome return in 1947 with the Vienna Staatsoper, singing the Commendatore in Don Giovanni and Rocco. When he rejoined the London company (now the Royal Opera House) for the 1949 - 1950 season, it was to dominate the Ring with his superlative singing and acting.
During his years in Munich and Vienna, Weber's Baron Ochs in Rosenkavalier ripened to a level second only to Richard Mayr's and his Boris was legendary. At the Salzburg Festival in 1947, he was part of a star-studded cast in the premiere of Gottfried von Einem's Dantons Tod. It was at Bayreuth, however, that Weber found artistic immortality, blending his eminence and experience with Wieland Wagner's vision to create unsurpassed realizations of Wagner's bass roles.
While Weber's top and bottom registers remained virtually unimpaired until his retirement, increasing unsteadiness and wayward intonation plagued his middle voice. These old-age failings, however, cannot detract from Weber's reputation as an artist of the highest ranking.