The Germans have agreed a fee of €23m for Bellingham, who is joining on a five-year deal.
The teenager is seen as a generational talent in England and the hefty price-tag reflects his potential.
Despite only turning 17 on June 29, Bellingham has made 35 appearances in the Championship this season.
The likes of Manchester United, Chelsea, Real Madrid and Bayern Munich coveted Bellingham over the last year.
United even gave him a tour of their Carrington training ground and were confident of winning the race.
However, he decided to follow in the footsteps of Jadon Sancho by playing at the Westfalenstadion next season.