The 23-year-old has penned a seven-year contract at Stamford Bridge and will replace Thibaut Courtois, who is set to join Real Madrid.
Kepa triggered his release clause earlier on Wednesday, which paved the way for his move away from the San Mames.
The fee eclipses the previous record amount that Liverpool paid Roma for Brazil custodian Alisson Becker last month.
"It's a very important decision for me, for my career, and also for my personal life," Kepa told the official Chelsea website.
"So many things attracted me to the club, all the titles the club has won, the other players, the city, the English Premier League.
"It's an accumulation of things, and I am very glad Chelsea has decided to trust me and to take me in as well."
Kepa is Spain's number two behind Manchester United's David de Gea and has one international cap.