Weller was able to read music before books as a child and at 18 was already a member of the Vienna Philharmonic. He made his conducting debut in 1966, stepping in for maestro Karl Böhm who was ill.
His career included stints at the Vienna State Opera, the Royal Liverpool Philharmonic Orchestra, the Royal Philharmonic Orchestra London, the Spanish National Orchestra and the Royal Scottish National Orchestra.
He was so popular at the Scottish National Orchestra that his face adorned a special edition of Scottish £50 notes. He also worked in Basel, Switzerland and was honorary conductor of the Stuttgart Philharmonic in Germany.