 Döme played as a captain for HC Slovan Bratislava in the Slovnaft Extraliga from the season 2007–08.Slovan Bratislava went on to win Slovnaft Extraliga that year.Döme played six games with HC Dukla Senica of Slovak 1.Liga, a second-tier league below Slovnaft Extraliga, during the 2009–10 season.