Goalkeeper De Gea, 20, passed a medical on Monday and has agreed a five-year deal worth £75,000-a-week.
"I feel very proud and I can't wait to start playing here," said De Gea, who will succeed Edwin van der Sar as United No.1.
"When a club the size of Manchester United comes in for you it obviously makes you very, very happy.
"I saw their interest as an extra motivation to work even harder to show off my ability.
De Gea told MUTV: "It's a great privilege to be part of a club like United and I'm keen to do my best and show what I can do."