Bergevin said on a video conference Thursday that Weber is dealing with a number of injuries, including to his foot, ankle and knee.
Weber, 35, was a stalwart on Montreal's blue line last season, guiding the Canadiens to the Stanley Cup Finals, where they lost to Tampa Bay in five games.
Bergevin said it will be impossible for the Canadiens to replace Weber, but the club will look to fill his minutes through trades, free agency and existing players.
"We'll try our best, but I know deep down that we can never replace Shea Weber," the general manager said.
Bergevin added that he recently had an "emotional, deep conversation" with Weber.
"It was hard for Shea," Bergevin said. "That's all he knows. He's a hockey player to the core. It's really hit hard to realize that he can no longer perform the way he's expecting for him and for his teammates, and the pain he's in going through daily."
Weber has 224 goals and 365 assists in 1,038 regular-season games with Nashville and Montreal, and has 18 goals and 24 assists in 97 playoff appearances.
Last season, he had six goals and 13 assists in 48 regular-season games and a goal and five assists in 22 playoff games.