Ideye moves to The Hawthorns in a deal that is believed to be in the region of £10 million.
The fee surpasses the one West Brom paid to sign attacking midfielder Stephane Sessegnon from Sunderland in September last year, when the Midlands club spent approximately £6m on the Benin playmaker.
West Brom's fifth signing of the transfer window, the 25-year-old has followed Joleon Lescott, Craig Gardner, Chris Baird and Sebastien Pocognoli to the club.
Head coach Alan Irvine told the club's official website: "Brown is a quality striker and I'm looking forward to working with him.
"He's a strong, quick, powerful player who likes to get in behind defences and has plenty of Champions League and international experience."
The former Sochaux man also has 23 Nigeria caps to his name, scoring five goals, although he was not part of the squad that reached the last 16 at the recent World Cup in Brazil.
