Costantin started in Brusson in December 1996 for the first time in the cross-country skiing world cup and finished 93rd place over 15 km freestyle. He ran his first race in the Continental Cup in February 1997 in Schlinig , which he finished in 14th place over 10 km freestyle. In February 2000 he clinched his only classic win in the Continental Cup in Monte Bondone over 15 km. In December 2000 he won his first World Cup points in Santa Caterina with 23rd place over 15 km freestyle. In the 2001/02 season he came in the Marathon Cup with four top ten placements in sixth place in the overall standings. He was fourth in the Tartu Maraton and the American Birkebeiner . In January 2003 he reached in Otepää with the 17th place in the mass start race over 30 km classic his best place in the World Cup. He repeated this result in February 2004 in La Clusaz over 15 km freestyle. At the Nordic World Ski Championships in 2003 in Val di Fiemme , he won 27th place in the 30 km mass start race. He also won the Toblach – Cortina in 2003 . In the 2004/05 season he came in the Marathon Cup six times in the top ten. He finished second in the American Birkebeiner and fourth in the overall standings at the end of the season. In the following season he reached seven top ten positions in the Marathon Cup, including third place each in the American Birkebeiner and the Transjurassienne . He finished second in the overall ranking. His last World Cup races he ran in the Tour de Ski 2006/07 , which he finished in 51st place.