"FC Nantes, Chelsea and Papy Djilobodji have reached an agreement for the transfer of the Senegalese international. Papy Djilobodji is a Blues player for the next four seasons," Nantes announced on their website.
Chelsea made their move for Djilobodji after failing in their protracted attempts to sign England international John Stones from Everton.
Djilobodji, who has been a regular in the Nantes side since signing in 2010 and helped them win promotion to Ligue 1 in 2013, will provide competition in central defence to John Terry, Gary Cahill and Kurt Zouma.
Chelsea's swoop for the unheralded Djilobodji was greeted with surprise in some quarters, with reports in England claiming Norwich, Bournemouth and Sunderland had all rejected the chance to sign him when he was offered to them earlier in the current transfer window.
But Blues boss Jose Mourinho sanctioned the surprise move because he was keen to secure defensive reinforcements after watching Chelsea leak nine goals in four games during a dismal start to their Premier League title defence.
How much Djilobodji can do to improve Chelsea's fortunes remains to be seen, but the player himself can't wait to get started at Stamford Bridge.
"I'm very happy to sign for a big club like Chelsea and I want to do my best to help the team win titles," Djilobodji told Chelsea's website. "I want to thank the club for giving me this opportunity."
Djilobodji is Chelsea's fifth significant signing since the end of last season following the arrivals of Pedro Rodriguez, Asmir Begovic, Baba Rahman and Radamel Falcao.
Mourinho added a sixth capture just before the window closed when he paid a reported £4m for Reading defender Michael Hector.
The 23-year-old centre-back has signed a five-year contract with Chelsea, but will return to Reading and spend the remainder of this season on loan with the Championship club.
London-born Hector, a boyhood Chelsea supporter, made 49 appearances for Reading in all competitions last season and helped Steve Clarke's team reach the FA Cup semi-finals.
He has represented Jamaica at international level and started all three of their matches at this year's Copa America.