Joe Allen, 26, has put pen to paper on a five-year contract after the clubs agreed a fee which Press Association Sport understands is £13million.
He has struggled for playing time under Jurgen Klopp at Anfield, having been restricted to mainly Europa League appearances last season.
But he was one of the stars of the Wales squad which reached the semi-finals at Euro 2016 earlier this summer.
Stoke’s bid met Liverpool’s valuation of a player with less than 12 months remaining on his contract and means they recouped most of the £15million they paid Swansea four years ago.
Allen told the club’s official website: “It’s been a whirlwind 48 hours for me, but I am delighted to have finally been able to put pen to paper.
“I was really excited as soon as I heard of the club’s interest and it wasn’t a difficult decision for me to make. I am incredibly excited about my future here at Stoke City.”
BREAKING | Stoke City are delighted to announce the signing of Joe Allen from @LFC on a 5-year deal
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8:00 PM – 25 Jul 2016
Stoke City FC
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Allen will fly out to link up with his new team-mates in Orlando on Tuesday after becoming manager Mark Hughes’ second summer signing, a few hours after Egypt winger Ramadan Sobhi arrived from Al Ahly.
Potters chief executive Tony Scholes added: “Joe is a player that Mark has admired for a considerable amount of time.
“Therefore when we became aware of the possibility of bringing him to the club it was something that we were determined to do.
“We have had to act swiftly to conclude the deal but we are delighted that we have managed to get it over the line.”
Allen joins winger Sobhi at the Britannia Stadium after Stoke agreed a fee which could rise to six million euros (£5million) for the 19-year-old, who has been capped six times by his country.
Scholes said: “We are absolutely delighted to have signed a player of Ramadan’s undoubted quality and potential.
“He’s a special young talent who is excited about the prospect of making an impact in the Premier League and we’re looking forward to giving him that platform.”
Sobhi has been granted a work permit and is expected to receive his UK visa in the next 48 hours. He should then join his new team-mates in Orlando later this week.
He is a graduate of the Al Ahly youth system and made his professional debut as a 16-year-old, earning his first international cap a year later.
He helped Al Ahly win the Egyptian League title last season under former Tottenham and Fulham manager Martin Jol.
Stoke have also tabled a £17million bid, rising to £20million with add-ons, for West Brom’s Saido Berahino.
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