Xherdan Shaqiri would not turn down the chance to play for Juventus after he was linked with the Italian champions.
Sassuolo have signed Croatia full-back Sime Vrsaljko from Genoa for a reported seven million euro fee.
The 20-year-old, part of Germany's World Cup-winning squad, has penned a five-year contract with the Westphalians.
"Matthias is a technically strong and versatile top German talent who has always enjoyed a high degree of interest from us," said Dortmund's director of sport Michael Zorc.
"We're pleased and also a bit proud that he made it clear so soon and so emphatically that he wanted to move to Borussia Dortmund.
"We are playing in three competitions, are very ambitious and have made a conscious decision to go into the season with four strong central defenders."
For Ginter, the move is something he had longed for his whole life.
"Today, I'm fulfilling a childhood dream," he said. "I was always a huge Dortmund fan and now I am getting the chance to wear the shirt of my favourite club.
"I would like to take the next step of my sporting development in Dortmund and I'm already looking forward to the amazing atmosphere in the Signal Iduna Park and being able to be part of this wonderful team after my post-World Cup holiday.
"I have only seen and heard positive things about this club."
Ginter will provide extra competition to a defence which was hit by injuries last season.
Neven Subotic is still recovering from a serious knee injury while Mats Hummels also spent some time in the sidelines last season.
Former Werder Bremen defender Sokratis Papastathopoulos is the fourth member of Jurgen Klopp's defence going into the new season.