The deal announced Thursday gives Dallas depth after forward Brad Richards broke his right wrist. Richards could be out two months.
The 30-year-old Begin has six goals and four assists in 42 games with the Canadiens. He has 84 points and 467 penalty minutes in 389 games.
"He's good on the penalty kill and he is a competitor in every sense of the word," Stars co-general manager Les Jackson said.
Janik played sparingly for Dallas this season, getting one assist in 13 games. He last played Dec. 20.