Weber, 30, had 20 goals and 51 points in 78 regular-season games, and three goals and seven points in 14 Stanley Cup Playoff games. He was the Predators captain.
"We completed today an important transaction which I am convinced will make the Canadiens a better team," Canadiens general manager Marc Bergevin said. "It was also one of the most difficult decisions I had to make as general manager of the Montreal Canadiens. In Shea Weber we get a top-rated NHL defenseman with tremendous leadership and a player who will improve our defensive group as well as our power play for many years to come. Shea Weber led all NHL defensemen last season with 14 power-play goals. He is a complete rearguard with impressive size and a powerful shot. P.K. Subban is a special and very talented player. He provided the Canadiens organization with strong performances on the ice and generous commitment in the community. I wish him the best of luck with the Predators."
A second-round pick (No. 49) in the 2003 draft, Weber is second in Predators history in goals (166) and games played (763), and third with 443 points. He has played in the NHL All-Star Game four times. He finished second in voting for the Norris Trophy in 2011 and 2012, and won the Mark Messier Leadership Award this season.
"I understand because I go back to the passionate fans and I understand you guys see P.K. everyday and he's been here for a while. He's a likable person. But when you look at Shea Weber I think you'll really get to appreciate [him]," Bergevin said. "When you see him twice a year or don't see him as much you really don't appreciate the type of defenseman he is. He's top notch.
"We didn't give P.K. Subban away. We got a very, very good player in return."
Weber has 10 years remaining on his 14-year, $110 million contract. He signed it with the Philadelphia Flyers as a restricted free agent but the Predators matched the offer sheet July 24, 2012.