Blais, 23, suffered the injury during the team's 3-1 win over the Tampa Bay Lightning on Tuesday.
Blais has eight points (five goals, three assists) and 10 penalty minutes in 20 games this season, his third with the team.
With Blais out long term, Alex Steen sidelined with a high ankle sprain until early December and Vladimir Tarasenko expected to miss the rest of the regular season following shoulder surgery, the team on Wednesday signed winger Troy Brouwer to a $750,000, one-year deal.
The 34-year-old Brouwer, who had been practicing with the Blues on a professional tryout, was the team's choice to sign over Jamie McGinn, who was also brought in for a tryout. A 13-year NHL veteran, Brouwer rejoins the Blues after playing the 2015-16 season in St. Louis.
The Associated Press contributed to this report.