Born in Odense, Helveg started his senior career at local club OB in the top-flight Danish 1st Division.
He made his Udinese debut in a 0–0 draw with Atalanta on 28 November 1993, and played 22 games and scored two goals in his first season at the club.
He came on as a substitute in the 14th minute for the injured John Jensen, and Helveg would receive rave reviews for his performance,[3] making him a mainstay in the Denmark squad from day one.
Helveg scored his first national team goal in June 1996, before representing Denmark in his first major international tournament at the 1996 European Championship (Euro 1996) later that month.
He played all three Denmark matches in the preliminary group stage of the Euro 1996, which ended unsuccessfully in an early exit from the tournament.
[4] Helveg played 30 of 34 games as Udinese finished in fifth place of the 1996–97 Serie A season, and qualified for European football for the first time in club history.
Both Helveg and Martin Jørgensen were included in new national manager Bo Johansson's Danish squad for the 1998 FIFA World Cup.
Their connection was evident in their joint celebration following Helveg's goal in Denmark's 4–1 win over Nigeria in the World Cup round of 16.
While the Euro 2000 was no success for neither Denmark nor Helveg, who had an injury which limited him to playing a combined 98 minutes in two games, the 2002 World Cup gave him more time on the pitch.
He played full-time in the three successful group stage matches, until another injury forced him to leave the pitch in the seventh minute of Denmark's final game before elimination.
Norwich had been promoted to the English FA Premier League and saw Helveg as a signing who would bring class and experience to the side.
His signing was seen as something of a coup for Norwich, but Helveg struggled to reproduce the form shown in Serie A and on the international scene, and he lost his place in the starting line-up in the opening weeks of the season.
After a long absence, partly due to injury, he regained his place in the side in early 2005, this time in central midfield.
After retiring at the end of 2010, it was confirmed on 17 December 2010, that Helveg from 1 January 2011 would continue at OB as an assistant coach under Henrik Clausen.