The former Borussia Monchengladbach is joining Barca in a Bosman.
"You don't get an offer like this every day. Barça is one of the biggest clubs in the world. They play exciting football and have had great success," he said.
"It is one of the best things that can happen to you and I will do everything I can to perform well. Coming to this club is a dream come true. I've taken a step forward."
Ter Stegen will replace departing Barca keeper Victor Valdes.
He added: "He is an extraordinary goalkeeper. He has done a lot for this club, won titles, performed very well but I'm focusing on my job."
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