He has played three seasons in the NHL, recording four goals and 37 assists for a total of 41 points and 48 penalty minutes in 128 career games.
The Baar, Switzerland, native appeared in the 2012 All-Star Game skills competition as a member of the rookie team.
Weise, 25, has collected three goals and nine assists for 12 points in 44 games this season.
"A young veteran with 162 games of experience at the NHL level, Dale Weise is a winger with a physical style of play who brings additional toughness and grit to our group of forwards," said Canadiens GM Marc Bergevin.
Weiss has registered 26 points (10 goals, 16 assists) and 185 penalty minutes in 162 NHL career games with the New York Rangers and the Canucks.
The Winnipeg native was the Rangers' fourth round selection, 111th overall, at the 2008 NHL Entry Draft.