Morris was born in Crickhowell, Wales, and graduated from Manchester Polytechnic. He made his England debut in 1988 against Australia. He was dropped in 1990 in favour of Richard Hill, but returned to the England side in 1992 and was the Lions scrum-half in 1993. Early on in his playing career, Dewi played for Winnington Park Rugby Football Club in Cheshire.
He retired from rugby union after the 1995 World Cup third place play-off match. He had won 26 caps in total. However, he came out of retirement after professionalism to play for a season for Sale, helping them reach the final of the Pilkington Cup. After retiring he now appears regularly as a rugby union pundit/commentator on Sky Sports where he appears with former England international (and Wales schoolboy international) Stuart Barnes and former England and Lions centre Will Greenwood.