Dann suffered the setback during the Carling Cup semi-final, first leg clash with West Ham at Upton Park.
The Blues centre-back underwent surgery on Tuesday and manager Alex McLeish's worst fears have been confirmed.
He said: "Scott had a successful operation the other day but he will be out for the rest of the season."
"You don't want to see your back four being disrupted too much.
"Scott has been an excellent performer, very consistent, and he is going to get even better.
"But I'm philosophical about these things. You've just got to get on with it.
"Managers can't legislate for injuries and suspensions and we've lost a good man.
"But it is up to his team-mates to show we won't miss him as much as we think."
Liam Ridgewell will partner Roger Johnson in defence but McLeish also has faith in Cech international Martin Jiranek if required.
He said: "We've got Martin Jiranek here as cover. Martin, although he is untried in the Premier League, is an experienced international. If something happens again, then Martin will come in."