The German defender has joined from SC Frieburg for €13m and becomes the Whites second signing of the day, after Rodrigo's arrival from Valencia was confirmed on Saturday morning.
The 24-year-old completed a medical on Saturday afternoon and is a former FC Kaiserslautern team-mate of United midfielder Mateusz Klich.
Leeds targeted Koch to strengthen their central defence, after Ben White returned to his parent club Brighton and Hove Albion at the conclusion of the 2019-20 campaign.
"I'm just very happy, everything is clear now," said Koch after penning a four-year deal at Leeds.
"It's one of the biggest leagues in the world. Wherever you go on holiday everyone knows the Premier League. Now to play with Leeds in the Premier League is a big dream.
"I think this is the best step for me under [Marcelo] Bielsa in Leeds."
Koch arrived in the Bundesliga with Frieburg in 2016, making 87 appearances for the club and scoring five goals.
The defender is on international duty over the coming week and will join up with the Germany squad tomorrow for games against Spain and Switzerland.
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