Anelka will move to the Hawthorns after two years with Chinese Super League side Shanghai Shenhua, having spent time on loan at Juventus last season too.
The 34-year-old has had a hugely successful career since joining Paris Saint-Germain at the age of 16, having won 17 different trophies with five different clubs.
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Clarke told the club's official website: "We are missing a bit of firepower from last year – we're trying to put that right – and to kick off with the signing of Nicolas at the start of the summer is a great boost for everyone at the club.
"The sole focus at the club is trying to build on what we did last season. The more quality players you can bring to the club the better – and no-one can question Nicolas' quality.
"He's a player I have worked with before as I was at Chelsea when they signed him. He's got a great work ethic, looks after himself and is a consummate professional. If you add that to the obvious talent he’s got I think he can prove to be a really top signing for the club.
"His experience with some of the top clubs in England and Europe will stand the group in good stead. Hopefully he will be able to inspire the other players to even greater heights than we managed to achieve last year."