"The player will be tied to the club for the next five seasons, until June 30, 2024".
The Belgium worldwide joins Los Blancos on a five-year-deal after spending seven seasons at Stamford Bridge, scoring 110 goals in 352 games and winning six major honours.
His final act in Chelsea blue was to score twice and assist once in the 4-1 Europa League final thrashing of Arsenal in Baku.
"Chelsea fans will always be special to me and next season I will look for your results first", Hazard said. "I think the beauty of Eden is to see an old-fashioned player with such a modern talent".
After more than a year of speculation amid contractual impasses and repeated hints of an exit, Hazard joins Luka Jovic and Eder Militao as blockbuster Real acquisitions in the early stages of the summer transfer window.
Mendy, who has two caps for France, has travelled with his national team for Saturday's Euro 2020 qualifier in Turkey, and will also be hopeful of featuring in Tuesday's qualifier in Andorra.
He won two Premier League titles, two UEFA Europa League titles, one FA Cup title and one League Cup title.
The Stamford Bridge hierarchy want their eventual new manager to arrive with the potential to make new recruits, with Barcelona's Philippe Coutinho a possible target as Hazard's replacement.
Whether it's on tour in the United States of America, or at Sunderland on a Tuesday night, in Russian Federation on a Thursday, or of course EVERY game at The Bridge - you've always supported me and encouraged me to play football and be me - Like I have enjoyed myself I hope you have enjoyed yourselves, I am only sorry not leave with a defending award amongst my honours!
Chelsea will now hope for success in their appeal to the Court for Arbitration for Sport (CAS) against their two-window Federation Internationale de Football Association transfer ban.
Hazard has been a long-term target for Madrid and hinted after last year's World Cup he planned to leave Chelsea.