The 23-year-old has signed a contract until 2021.
Ginter was part of the Germany side that beat Chile 1-0 in Sunday's Confederations Cup final.
Dortmund's sporting director Michael Zorc said Ginter had "expressed his desire to leave" because of competition for places in the Dortmund squad.
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Borussia Monchengladbach have signed Germany defender Matthias Ginter from Borussia Dortmund for a reported 17m euros (£14.9m) plus add-ons.