Weber had his ankle re-evaluated Wednesday to determine the severity of the injury, and a full recovery is expected.
Montreal's captain suffered the injury Feb. 4 against the New Jersey Devils. He was placed on injured reserve two days later.
Weber leads the Canadiens in ice time with an average of 24:18 per game and has contributed 13 goals and 34 points through 55 games this season.
The veteran's absence is a crucial blow to Montreal's playoff push. The team sits fifth in the Atlantic Division with 61 points and is currently eight points back of a wild-card spot.
Weber has battled injuries for much of his tenure with the Canadiens, missing 80 games between the 2017-18 and 2018-19 seasons.