Garoppolo was a second-round pick in this year's draft - the highest coach Bill Belichick has ever selected a quarterback in New England.
''We have three good players at that position, three players we have a lot of confidence in,'' Belichick said on Sunday just hours before the trade was announced. ''It's a good situation.''
Mallett was 14 for 26 for 161 yards and a touchdown this preseason, and his fate seemed decided when Garoppolo played the entire exhibition finale on Thursday night against the New York Giants.
Originally a third-round draft pick from Arkansas, Mallett played four regular-season games in three years with the Patriots, completing one pass.
In Houston, Mallett will move into the backup role created when the Texans cut Case Keenum later Sunday. Keenum had been competing with rookie fourth-round pick Tom Savage for the No. 2 role behind Harvard product Ryan Fitzpatrick.
In the preseason, Garoppolo was 46 for 79 for 618 yards, five touchdowns and an interception.
But the Patriots are hoping he returns to the bench and stays there.
Brady has only missed games because of injury once in his career since inheriting the starting job from Drew Bledsoe in 2001. With Brady's knee torn up, Matt Cassel led the Patriots to an 11-5 record in 2008 but missed the playoffs - one of just two seasons in the Brady era in which New England failed to make the postseason.
Also Sunday, the Patriots claimed defensive linemen Bruce Gaston and Kelcy Quarles off waivers. New England released linebacker Chris White.
Gaston was a rookie free agent from Purdue cut by Arizona on Saturday. He played 50 consecutive games at Purdue and finished with 131 total tackles, seven sacks, four fumble recoveries and two forced fumbles.
Quarles was cut by the Giants on Saturday. He played in 35 games in three years at South Carolina, finishing with 105 total tackles and 13 sacks.
White played two seasons with the Buffalo Bills before joining the Patriots last season. Originally a sixth-round draft pick from Mississippi State, he played in 16 games and finished with one tackle on defense and nine special teams tackles.
AP Sports Writer Chris Duncan contributed to this story from Houston.
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