The 24-year old has completed the signing of a three-year deal after he passed a medical, making him the second new player to join the Baggies so far this off-season.
"I'm delighted to welcome Billy to the club," he said.
"We're really pleased we've been able to secure his signature in the face of stiff competition from other clubs for his services.
"It is great that he has chosen our club as the place to further his career and we are looking forward to working with him.
"We have been monitoring Billy's progress for some time and, like the player himself, we believe this is the right moment for him to move up a level.
"He has been a very consistent performer in the Championship for a long period and this is his chance to show he can be as good a player in the Premier League."