The 25-year-old Spanish international has shone on loan in Germany this season, scoring nine goals in eight games for Dortmund, prompting the Bundesliga leaders to make the move permanent with a five-year deal from January.
"Today we informed Barcelona we will activate the option to buy," Dortmund's sporting director Michael Zorc said in a statement on Friday.
Having struggled to break into the Barcelona side after his move from Valencia in 2016, Alcacer found a new home in Dortmund and was recently recalled to the Spain squad thanks to his good form for the German club.
"I am very happy in Dortmund, at the club and in this wonderful squad," said Alcacer.
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"Borussia Dortmund has been and is exactly the right decision for me. I will do everything I can to pay back their trust with performances and goals."
(Reporting by Rik Sharma; Editing by David Goodman)
