Ceci exited Wednesday's contest versus the New York Rangers in the third period and didn't return.
Toronto is also without top defenseman Morgan Rielly, who was ruled out for eight weeks on Jan. 13 due to a broken foot.
Ceci has suited up in all 54 games for the Maple Leafs this season, posting one goal and seven assists while averaging more than 20 minutes of ice time per night. Toronto acquired him in the offseason from the Ottawa Senators.
Weber suffered the injury during Tuesday's victory over the New Jersey Devils, head coach Claude Julien confirmed, according to Sportsnet's Eric Engels.
The 34-year-old will be out of the lineup for at least a week.
Montreal recalled defenseman Xavier Ouellet from the AHL's Laval Rockets on Thursday, the team announced.
Weber leads the Canadiens in ice time with an average of 24:18 per game. He also ranks second on the club in blocks and has contributed 13 goals and 34 points through 55 games.
The Canadiens sit seven points shy of third place in the Atlantic Division and eight points back of the second Eastern Conference wild-card spot with 27 games remaining.
Anthony Mantha is nearing a return to the lineup, but it turns out that his ailments were worse than they initially appeared.
In addition to hurting his ribs, the Detroit Red Wings forward revealed Wednesday that he suffered a second significant injury when Toronto Maple Leafs defenseman Jake Muzzin slammed him to the ice Dec. 21.
"I had a punctured lung," he told reporters, including the Detroit Free Press' Helene St. James. "That's what took the six weeks to heal. We didn't want to risk that. Then obviously the ribs, it's a matter of time before it heals. And then the head was just preventative."
To make matters worse, Mantha also came down with an illness shortly thereafter.
"With the force of impact, my ribs punctured my lungs," he added. "It was probably the worst pain I've been in. And on top of all that, I got the flu on Day 2 of being injured. I had (a) 103 (degree) fever. So it was just a terrible first week."
Mantha passed a final concussion test Tuesday morning.
"I don't think I had a concussion, but we just wanted to make sure," he said.
The 25-year-old has missed 17 games since the incident but said he hopes to return between Feb. 10 and 15. The Red Wings will be on the road in Buffalo (Feb. 11), New Jersey (Feb. 13), and Boston (Feb. 15) during that span.
Mantha poured in 12 goals and 24 points across 29 games before getting hurt and still leads the club in points per game this season.
Ottawa Senators forward Bobby Ryan practiced Wednesday for the first time since entering the NHL/NHLPA player assistance program in November, the team announced.
The veteran winger, who hasn't played since Nov. 16, was medically cleared and is working on getting his conditioning back.
"It's like missing training camp ... he is a ways away but he took his first step," head coach D.J. Smith said, according to TSN's Brent Wallace.
There is no timeline for Ryan's potential return to the lineup.
Ryan, 32, is the Senators' highest-paid player with an annual cap hit of $7.25 million. He has two more years beyond this season remaining on his contract.
The New Jersey native recorded one goal and four points through 16 games before stepping away from the team.