The Potters have paid £1.3million for the 33-year-old goalkeeper who had been on loan at the bet365 Stadium since August.
Despite being a newcomer to top flight football, Grant has been one of the revelations of the Premier League season and has a signed a two-and-a-half year contact.
The former England Under-21 international has made 15 league appearances and kept six clean sheets, winning a number of plaudits along the way.
City Chief Executive Tony Scholes said: "Lee found himself with an opportunity to make an impression in the Premier League and to suggest he has grabbed that chance with both hands would be an understatement.
"He has shown outstanding form on a consistent basis and we're delighted to have reached an agreement with Derby to make his transfer permanent."