Ndiaye, who only joined Galatasaray in August for around £6.5m, has put pen to paper on a four-and-a-half year contract that promises to keep him at the bet365 Stadium until 2022.
Speaking to the club's official website, the Sengalese midfielder expressed his delight at joining a "very big club in England':
“I am very happy to join Stoke City, a very big club in England - it is a proud moment for me,” he stated.
“There have been many talks this month but I wanted to join Stoke City, so to be here now is very exciting for me and I am looking forward to helping the team move forward.
“I knew many good things about the Club, I spoke to friends who play in the Premier League and I also spoke to Mame Diouf, who is a very good friend of mine.
“They were all very positive and made me very excited to make this move, so as I have said before I am just very happy to be here now.”
Prior to joining Galatasaray, Ndiaye was already playing in Turkey with Osmanlispor, with whom he enjoyed an impressive return of a goal almost every three games during two seasons there. The 27-year-old earlier played in Norway after arriving in Europe from Senegal in 2013.
Ndiaye has been a full Senegal international since 2014 and was part of the most recent squad that completed the 2018 World Cup qualifying campaign in November. He only has eight caps to his name, but looks to be on course to go to the World Cup in summer.
With Ndiaye joining earlier January signings Kostas Stafylidis and Moritz Bauer, Stoke are not now expected to do any further business before the 11pm deadline.