The 24-year-old has signed a four-year contract after establishing himself as the club's first-choice keeper during two seasons on loan at Celtic Park.
And Forster, who set a new SPL record of 25 clean sheets alongside back-up shot-stopper Lukasz Zaluska last season, expressed delight at sealing the move.
"This is a massive club and I feel like this is a step forward in my career," Forster told the club's official website. "I want to win things as a team and improve as an individual and I think I will be able to do that at Celtic.
"It also gives me a great opportunity to play in the Champions League if we get through the qualifiers. I really enjoyed my time in Europe last season so I will be looking forward to it if it happens again.
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"I enjoyed last season as a whole and we have a great young squad here. The players and the coaches make it an enjoyable place to be and I like being a part of this team."
Celtic manager Neil Lennon also hailed Forster's signing ahead of a new SPL season in which the Bhoys are strong favourites to defend their title.
"We are delighted that Fraser has committed his future to the club and we believe he will be a magnificent addition to the squad," Lennon enthused.
"He gives us that continuity so this is something we are really pleased about. He is a top-class keeper."