The Stars will receive a 2021 third-round pick in the deal.
Dickinson, 26, has spent his entire NHL career in Dallas. Last season, he recorded seven goals and 15 points over 51 games.
Over 221 career NHL games, the centre has posted 25 goals and 63 points.
The Georgetown, Ont., native was the No. 29 pick in the 2013 NHL Draft.