He began recording in the early 1960s, and released his first record, Colours of Chloë (ECM 1042), under his own name in 1973. In addition to his career as a musician, he also worked for many years as a television and theater director. He has designed an electric-acoustic bass featuring an extra C-string.
His music, often in a melancholic tone, follows simple ground patterns (frequently ostinatos), yet is highly organized in its colouring and attention to dramatic detail.
Weber was a notable early proponent of the solid-body electric double bass, which he has played regularly since the beginning of the 1970s.
From the early 1960s to the early 1970s, his closest musical association was with pianist Wolfgang Dauner. Their many mutual projects were very diverse, from mainstream jazz to jazz-rock fusion to avant-garde sound experiments. During this period he also played and recorded with (among many others) pianists Hampton Hawes and Mal Waldron, guitarists Baden Powell de Aquino and Joe Pass, and violinist Stephane Grappelli.
In 1973 he made his first record as a leader, The Colours of Chloë, for ECM Records. Since then he has released 10 more records under his own name, all on ECM. The ECM association also led to collaborations with other ECM recording artists such as Gary Burton (Ring, 1974; Passengers, 1976), Ralph Towner (Solstice, 1974; Sound and Shadows, 1977), Pat Metheny (Watercolors, 1977), and Jan Garbarek (nine recordings between 1978 and 1998).