Rosenberg, who had previously played for Malmo, Ajax and Racing Santander, has been on the Baggies radar since it became clear his days at Bremen were numbered.
He held talks yesterday afternoon with Albion at their training ground and passed a medical before agreeing to sign subject to international clearance.
Clarke said: "Markus is an experienced striker, has scored goals in three of Europe's top leagues and top European competitions, and is determined to prove himself in the Premier League.
"He will provide real competition for places in the forward areas and hopefully spur on the strikers who are already here.
"Markus has had several other offers this summer but made it clear joining West Brom was his first choice."