The German, 24, will join the Canaries on a two-year deal when his contract at Dortmund runs out on 1 July.
He played 29 times for Dortmund II last season under Norwich's new boss Daniel Farke who took over in May.
"I'm extremely delighted and very honoured for the opportunity. I'm already counting the days to get started," Zimmermann said.
"I've had a really positive first impression, so I'm looking forward to meeting all the guys and I cannot wait to start working."
Zimmermann joined Dortmund's reserve team in 2014 after two seasons with Borussia Monchengladbach II where he progressed through the youth teams.
Farke said: "He's very big, tall, strong and powerful and he's an intelligent boy.
"He was my captain of my under-23 team at Borussia Dortmund, and he fits English football because of his powerful way.
"We're very happy he's chosen to come to Norwich."
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Championship club Norwich City have signed Borussia Dortmund II central defender Christoph Zimmermann.