Financial terms were not disclosed by the team for Bradford, who spent the last two seasons with the Minnesota Vikings.
Bradford effectively replaces Carson Palmer after the latter announced his retirement in January. The 38-year-old Palmer was named to the Pro Bowl three times, but his 2017 season was cut short by a broken arm that limited him to seven games.
The 30-year-old Bradford performed well in his first season with the Minnesota Vikings in place of an injured Teddy Bridgewater, posting a then-NFL single-season record for completion percentage (71.6). He also set a Vikings franchise record with 395 completions and became the first player in league history to throw five-or-fewer interceptions while attempting 550-plus passes in a season.
Bradford was not as fortunate the following season, namely the 2017 season opener. He sustained a non-contact left knee injury during Minnesota's 29-19 victory over the New Orleans Saints.
The injury was to the same knee that Bradford had twice torn his ACL.
Bradford returned in Week 5 against the Chicago Bears, but he was pulled before halftime after aggravating the injury. He went on injured reserve and had arthroscopic surgery in November before being activated as Case Keenum's backup during the postseason.
A former Heisman Trophy winner and top overall pick of the 2010 NFL draft, Bradford has thrown for 19,049 yards with 101 touchdowns and 57 interceptions in 80 career contests with the then-St. Louis Rams, Philadelphia Eagles and Vikings.