Following the Russian invasion of Ukraine on February 24, owner Roman Abramovich was sanctioned for his close ties to Russian President Vladimir Putin.
Meanwhile, Chelsea have continued the business with a special license developed by the government.
It was to expire on 31 May.
"Late last night, the British government reached a point where we could issue a license that allows the sale of Chelsea," a government spokesman told the BBC.
It has been a long and intense process where it was recently said that the whole business was about to collapse.
The oligarch Abramovich has publicly denied that he would like to repay a loan of SEK 18 billion in connection with the sale.