The 26-year-old, who had a year left on his contract, was also a target for the Potters' Premier League rivals Swansea, whom he left
to
join Liverpool in 2012.
"It's been a whirlwind 48 hours for me but I am delighted to have finally been able to put pen to paper," said Allen, who has signed a five-year contract.
"I was really excited as soon as I heard of the club's interest and it wasn't a difficult decision to make."
Allen was Stoke's second signing of the day after
Egyptian winger Ramadan Sobhi joined the club from Al Ahly for a fee that could rise to £5m.
Despite starting only eight league games for Liverpool last season, Allen was named in the Euro 2016 team of the tournament after helping Wales reach the semi-finals.
Liverpool rejected an £8m offer from Swansea in June but Allen will now link up with Stoke's Welsh manager Mark Hughes.
"Joe is a player that Mark has admired for a considerable amount of time," said the club's chief executive Tony Scholes.
"We have had to act swiftly to conclude the deal but we are delighted that we have managed to get it over the line."