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.