The former Ipswich goalkeeper’s contract had expired and although the Portman Road club offered the Hungarian a new deal, the 28-year-old elected to move to the Midlands.
"I'm very happy that we've been able to address the issue of signing another goalkeeper so quickly," said West Brom manager Roy Hodgson.
"I'm also delighted that we've signed a player of Marton's quality and experience.
"Like last year, we've now got two goalkeepers with lots of Premier League experience who are both very capable of playing in the top flight and helping us to win matches.
"I'm really pleased the deal for Marton has gone through so quickly so that we have both keepers in place for the opening game of the season."
Fulop joined Town from Sunderland last season, making 35 appearances.