 After playing three seasons in the International Hockey League, Döme was drafted 17th overall in the 1997 NHL Entry Draft by the Pittsburgh Penguins.He appeared in 30 games with the Penguins in the 1997–98 season, and 22 more in the 1999–2000 season.He made his most recent NHL appearance in the 2002–03 season, playing in one game with the Calgary Flames.