The 28-year-old has joined on a one-year deal from Ipswich after passing a Hawthorns medical.
"I'm very happy that we've been able to address the issue of signing another goalkeeper so quickly," Hodgson told the WBA website.
"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."