Rudiger, who can play at centre-back or right-back, has signed a five-year contract and becomes Chelsea’s second signing of what is expected to be a busy summer.
His arrival brings defensive reinforcement to Antonio Conte’s squad after Nathan Ake joined Bournemouth and John Terry was released at the end of his contract.
Rudiger, 24, was heavily linked with a move to Chelsea last summer before tearing the cruciate ligaments in his right knee in training.
He told Chelsea’s official website: “It’s a great feeling because not every player has the opportunity to join a huge club like this.
“I am looking forward to meeting up with my new teammates and I’m very proud to officially become a Chelsea player.”
Technical director Michael Emenalo said: “Antonio is still young but is experienced at club and international level and possesses all the requisite attributes to thrive in the Premier League.
“We have been aware of him for some time, and we are confident he will fit in well with the squad.”
Stuttgart, the club he left to join Roma in 2015, stand to make a sizeable profit from his move to Chelsea, German football magazine kicker reported.
Chelsea began their summer recruitment with the acquisition of free agent goalkeeper Willy Caballero at the beginning of July, but suffered a setback when Manchester United beat them to the £75m signing of Romelu Lukaku from Everton.
A £35m deal for Monaco midfielder Tiemoue Bakayoko is expected to be confirmed in the coming days, while talks are ongoing over the possible £61m signing of Alex Sandro from Juventus.
Missing out on Lukaku, however, means the Premier League champions will be forced to look elsewhere for a replacement for Diego Costa, who remains likely to be sold to Atletico Madrid, with Alvaro Morata and Andrea Belotti high on Conte’s wish list.