During his active career, Wieghorst played as a midfielder. He started his career with Lyngby Boldklub, with whom he won the 1990 Danish Cup and 1992 Superliga titles. He then moved abroad and played 10 seasons in Scottish football with Dundee and Celtic, and won the 1998 Scottish football championship with Celtic. He ended his career with Danish club Brøndby IF, after winning the 2005 Superliga and two Danish Cup trophies with the team. He played 30 matches and scored three goals for the Denmark national football team from 1994 to 2004, won the 1995 Confederations Cup, and appeared in three games at the 1998 World Cup. He was named 2003 Danish "Player of the Year", and received a 2003 Olympic Committee fair play award for missing a penalty kick on purpose.