The 30-year old Begin has played in 42 games this season for the Canadiens, and recorded 6 goals and 4 assists. He is in his 9th season, having begun his career with the Calgary Flames before being picked up off waivers by the Canadiens prior to the 2003-04 season.
The native of Trois-Rivieres, Quebec had a career-high of 23 points in 2005-06.
"Steve is a gritty, honest player with a very strong work ethic," said Stars Co-General Manager Les Jackson in a statement. "He's good on the penalty kill and he is a competitor in every sense of the word. We think he'll be a very good addition to our group as we continue to fight toward the playoffs."
Janik, 28, has spent the majority of the season with Rockford of the AHL. He has one assist and three penalty minutes in 13 games this year with Dallas. He started his career with the Buffalo Sabres, and played last season with the Tampa Bay Lightning.
"Throughout his five seasons in the Montreal Canadiens' jersey Steve Begin has been a proud competitor who dedicated himself to the team," said Canadiens general manager Bob Gainey in a statement. "As an organization, we are grateful to Steve for his play on the ice as well as his involvement in the Quebec community. We wish Steve the very best with the Dallas Stars. Doug Janik, obtained by the Canadiens in return for Bégin, is a solid defenseman who brings 159 games of experience at the NHL level and who will add depth to our group of defencemen."