Highly-rated striker Brobbey, 19, has signed a four-year deal with the Bundesliga side after it was announced early hours of Friday morning by Leipzig.
The Netherlands U21 international has enjoyed a stunning breakthrough season for Ajax, scoring 13 goals, as well as providing five assists, from 26 games in all competitions.
In January, Real Madrid were eyeing the Dutch sensation but he has opted for the German side, who have a history of developing young talent.
He will give Julian Nagelsmann some much-needed competition in the forward positions following the departure of star striker Timo Werner last summer.
Brobbey scored two goals in seven league appearances for Ajax this season and two in three Europa League appearances.