The former England international, who arrived on loan from Tottenham Hotspur in August, was taken off on a stretcher during Villa's home game last Saturday against Manchester United.
The club said Jenas, who has hardly featured for Villa, was expected to be out for around six months.
"This is tragic for Jermaine and of course we are all deeply disappointed for him," said manager Alex McLeish in a statement on the club website.
"To lose a player of his talent and experience is bad enough but never to have seen him play enough in the Villa shirt for our fans to appreciate the player he is, that's terrible bad luck."
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