He suffered the injury during last Sunday's loss to the St. Louis Rams.
Brown, a second-round pick in the 2010 NFL Draft, started the last five games at right tackle with Zach Strief out with a knee injury. But the 28-year-old Strief returned to practice on Wednesday and is expected to start this Sunday against Tampa Bay.
To add depth on the offensive line, the Saints signed veteran tackle Ray Willis. The 29-year-old spent six seasons with the Seahawks, and started 26 games at right tackle and guard before missing all of last season with a knee injury.