The right-back,19, has agreed on a five-year contract with the Bundesliga club and arrives on a free transfer after running down his deal at Barca.
Morey, who was also reportedly a target for Juventus and Manchester City, missed the entire 2018-19 season after suffering a serious knee injury but is now fully fit and set to be involved in Dortmund's pre-season.
"Borussia Dortmund are a fantastic club that places a lot of emphasis on youth. I'm confident that, with my individual qualities, I will have a lot to offer the club in the years to come," Morey said.