The USA international has penned terms to 2025.
The 19-year-old was able to take part in his first training session for Ronald Koeman's side on Thursday afternoon.
"Everything I'm experiencing is incredible," Dest told Barcelona's official media channels.
"It is an honour to play for the best team in the world. It's a dream come true.
"I came here to get the chance to play with the best players in the world.
"But the first thing I want to do is improve, to play well, and then try to win everything I can."