Baird was allowed to come off contract by Fulham this month.
The 32-year-old has penned a one-year deal, which includes a further performance-related 12-month option, after agreeing personal terms and passing a medical.
WBA boss Alan Irvine said: "I'd like to officially welcome Chris to the club.
"Chris is a player with vast Premier League and international experience who reads the game well and has good positional sense
"He can play in several positions and his versatility effectively makes your squad bigger.
"It's important we have the right characters in the dressing room and Chris is certainly that.
"We have done our homework on him. I have spoken to a lot of different people who have either managed or played with Chris and everyone spoke very, very highly of him both as a player and a person."
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