The 25-year-old Republic of Ireland defender has joined the Potters on loan for the remainder of the season.
Kelly's move to The Britannia Stadium had been agreed between the two club's before the closure of the transfer window at 5pm on 2nd February but because of the adverse weather conditions across Britain on Monday, the FA and Football League gave club's special dispensation on completing signings as long as they received emails or faxes confirming that the deal had been agreed in principal.