Weller began learning clarinet at the age of 14, and was classically
educated on it for four or five years, soloing in Mozart's Clarinet
Concerto at Croydon Town Hall when aged 15. He began playing in
Dixieland bands around the Croydon area, but later switched to tenor
In the 1970s his jazz-rock group Major Surgery played only Weller's compositions. This was followed by a quartet with drummer Bryan Spring. At the same time he worked regularly with pianist Stan Tracey, and also with Harry Beckett and in a quintet with Art Themen. Renowned for his versatility, he has played with artists such as Alan Price, Tina May and Charlie Hearnshaw.
In 1980 he stood in for Michael Brecker when the Gil Evans Orchestra played at the Bracknell Jazz Festival, subsequently touring the UK and recording with one of the band's trumpeters, Hannibal Marvin Peterson, added on to the Weller–Spring Quartet. He also played and recorded with Evan's British Orchestra in 1983.
Also in the 1980s, together with his friend and fellow tenor sax Dick Morrissey, Weller was a regular member of Rocket 88, the boogie-woogie fun band set up by Ian "Stu" Stewart and Bob Hall, appearing on the band's only album.He also formed a band with fellow saxophonist Bobby Wellins which included Bryan Spring on drums and US pianist Errol Clarke. This partnership reunited for the Trio Records album "Nine Songs" TR577
He has been involved with the films Absolute Beginners and Stormy Monday both as performer and composer. He won the 'Top Tenor' award in 1994, 1996 and 1998. He formed his big band to perform his 'Pennine Suite' at the 1996 Appleby Jazz Festival, and since then the band has regularly appeared at other jazz festivals. He also formed an 'Electric Jazz Octet' to perform at the Appleby Jazz Festival.
Don's present quartet includes Dave Newton on piano, Andrew Cleyndert on bass and Dave Barry on drums which has also recorded for Trio Records.