The forward, who was on the United States provisional roster for the World Cup, will join the 2. Bundesliga club next season.
After playing there for two seasons, Boyd leaves with a year left on his contract with Rapid. The 23-year-old forward scored 23 goals in 59 league appearances for the club and scored eight times in 17 Europa League matches.
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Boyd made known his desire to move to a more prominent league than the Austrian Bundesliga and now returns to Germany, his country of birth. Terms of the deal have not been announced, but German tabloid Bild suggested last week that the 2. Bundesliga outfit could pay 1.5 million euros for Boyd's services.
Rapid sporting director Andreas Muller said the club wanted to keep Boyd long-term, but elected to sell him when the forward declined and wished him success in a news release.
The player also gave a statement in the news release, saying that he valued his time at the club and regretted not helping the team to a title.
Boyd, born to an American father and a German mother, has more than a dozen senior U.S. caps and made Jurgen Klinsmann's preliminary World Cup roster before being one of seven players cut from the team.