The 36-year-old defenceman led the Canadiens to the Stanley Cup Final in 2021, but missed all of the 2021-22 season due to various injuries.
Weber was drafted 49th overall by Nashville in the 2003 NHL entry draft.
The Predators dealt him to the Habs for defenceman P.K. Subban in a blockbuster deal ahead of free agency in June 2019.
Weber, who hails from Sicamous, B.C., has played 1,038 regular-season NHL games, putting up 589 points (224 goals, 365 assists).
Dadonov, 33, played 78 games with the Golden Knights in 2021-22, recording 20 goals and 23 assists.
“First of all, I would like to thank Shea for all that he has done for and represented to the Montreal Canadiens,” Canadiens general manager Kent Hughes said in a statement.
“Not only was Shea an exceptional hockey player who played through significant injuries in an attempt to bring the Stanley Cup back to Montreal, he was also a great leader whose leadership will have a positive impact on our team well beyond his time with us.”