The 24-year-old joins from Norwegian side Rosenborg and has signed a three-year contract with Alan Irvine's side after his work permit application appeal was accepted
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“As I commented the other day, it would have been amazing if Cristian had not been granted a work permit and we’re obviously satisfied with today’s outcome," Irvine told the club's official website.
“Cristian is a player I was extremely impressed by during the World Cup.
"We faced stiff competition from other Premier League clubs for his services but we've managed to convince Cristian that West Bromwich Albion is the best club for him to join.
"Cristian is a quick, aggressive defender with good energy levels. He gets up and down the pitch really well and we believe he will continue to improve."
Gamboa becomes West Brom's eighth signing this summer following the arrivals of Brown Ideye, Craig Gardner, Joleon Lescott, Chris Baird, Sebastien Pocognoli, Jason Davidson and Andre Wisdom, who has joined on a season-long loan from Liverpool.