Imbula, 23, has signed a five-and-a-half year contract, said the club's website (www.stokecityfc.com).
The fee dwarfs the 12 million pounds that the Premier League club paid for Swiss forward Xherdan Shaqiri from Inter Milan in August.
"I'm delighted. It was a long day but we have got a very good young player who has a lot of potential," Stoke manager Mark Hughes told Sky Sports.
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"A lot of things had to fall into place but they did thankfully."
Imbula has Champions League experience with Olympique Marseille, the club he joined in 2013 from French rivals Guingamp.
He moved to Porto last year for 20 million euros ($21.77 million).
Stoke chief executive Tony Scholes said: "Giannelli is a player we have been trailing for some time.
"At one stage it looked unlikely we would be able to get the deal over the line but things accelerated quickly earlier today and Giannelli is really excited about the prospect of playing in the Premier League with us."
Stoke, who are ninth in the table, visit fifth-placed Manchester United on Tuesday.
(Writing by Martyn Herman; Editing by Tony Jimenez)
