The former Birmingham City forward arrived on transfer deadline day.
"When you see players like Peter joining the Club, you soon realise the ambition of the manager and the Chairman, and I soon decided I wanted to be a part of that," Jerome told Stoke's website.
"These are exciting times at Stoke and it's a massive thing to be part of that. I am really delighted that I have signed and I am looking forward to the challenge facing me here.
"The most important thing from my point of view was getting back into the Premier League but another key factor was European football which will be a new experience for me."