The Spain attacker's move from RB Leipzig was confirmed earlier on Friday.
Olmo, a former La Masia prospect, joins for €55m up from plus a further €7m due in bonuses.
He has signed a contract to 2030.
Olmo said: " I'm very happy, it's a dream come true, I'm returning home with my family and teammates. I'm looking forward to getting started, I'm very proud that Barça has made this effort, that they love me."
On his strengths, Olmo continued: "Energy, quality, first touch, I have already played with many Barça players. There is a lot of quality here, I want to play with players like (Robert) Lewandowski and I am prepared to contribute.
"The objective is to win, we are Barça and we have to compete for everything, we have a team, we have players, we have staff and we are going to go for everything."