The team said Tuesday the all-star defenceman suffered the injury in Friday's 6-3 loss to the Dallas Stars. There is no timetable for his return.
Weber joins a long list of high-profile players currently out with concussions, including Pittsburgh's Sidney Crosby and Carolina's Jeff Skinner.
The Philadelphia Flyers announced earlier this month captain Chris Pronger will miss the rest of the 2011-12 campaign with a concussion.
Weber has eight goals and 21 assists in 35 games for Nashville this season.
Joining Weber on the sidelines is Toronto Maple Leafs defenceman John-Michael Liles. He's out with concussion-like symptoms.
Liles took a big hit from Buffalo's Paul Gaustad in Thursday's 3-2 win over the Sabres.
The defenceman has four goals and 17 assists in 34 games with the Leafs this season.