The 25-year-old has joined from Dynamo Kyiv, where he scored 33 goals in 74 games for the Ukrainian outfit.
He has signed a three-year contract and has become the club's record signing.
"Brown is a quality striker and I'm looking forward to working with him," West Brom manager Alan Irvine told the club's official website.
"He's a strong, quick, powerful player who likes to get in behind defences and has plenty of Champions League and international experience."
Ideye has made 23 appearances for Nigeria, but he was not part of their World Cup squad this summer.