The 29-year-old Branch spent the 2013 season with the Buffalo Bills, appearing in all 16 games and registering 39 tackles. He was released by the team in late August after an arrest for DUI.
The Patriots released rookie linebacker Deontae Skinner to make room for Branch on the roster.
Branch is a former second round draft pick by the Arizona Cardinals in 2007. He spent four seasons there before signing with the Seattle Seahawks, where he played for two years before going to Buffalo.
In his NFL career, Branch has played in 94 games (47 starts) and has 170 tackles and eight sacks.
Skinner signed with New England in May as an undrafted rookie free agent. He played in seven games, totaling 10 tackles.
The Patriots (6-2) host the Denver Broncos (6-1) on Sunday.
- Brad Rowland