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Watt confirmed his decision to sign for the Cardinals in a post on Twitter showing himself wearing an Arizona training top.
Multiple reports said the 31-year-old had signed a two-year deal worth $31 million.
Following his release from Houston last month, Watt had been linked strongly with several teams seen as more viable championship contenders including the Green Bay Packers, Buffalo Bills and Cleveland Browns.
In joining Arizona he will be arriving at a franchise that has failed to make the playoffs for the past five seasons from the NFC West, arguably the most competitive regional division in the NFL.
Watt has been a first-team All-Pro selection five times in his 10 NFL seasons and twice led the league in sacks, with 101 career sacks for the Texans.
He won Defensive Player of the Year honors in 2012, 2014 and 2015.
He became a hero in Houston in 2017, helping the city's recovery from Hurricane Harvey by raising more than $40 million for relief efforts.