Carter-Vickers joined Celtic on a four year deal.
The USA international spent last season on loan in Glasgow and helped the side win the Scottish title and League Cup.
Carter-Vickers made 45 appearances for Celtic and scored four goals including one on his debut against Ross County and a key winner against Rangers.
He said he was happy to complete his move to Celtic.
I've loved my time at Celtic so far and I wanted to be a part of the future.
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There is a good group of players at the club, we have a good spirit and atmosphere, and we all worked so hard to deliver the league and League Cup success last season."
We all want to achieve more and I wanted to make sure I was a part of that.
"We have so much to look forward to, we will aim to be strong again domestically and of course we look ahead with real excitement to the champions league."
I have been welcomed by everyone at the club and our supporters. You can see that Celtic is a great football club when you are inside.
I am very happy to be given this opportunity and I will do my best to help bring more and more success to our fans.
The manager said he was delighted that the club had completed the signing of Cameron on a permanent basis.
This is great news for Celtic and everyone at the club has worked so hard to make it happen.
He has shown his value to the team and the way we play by giving everything of himself to the club so far.
He was an absolute model of consistency and was vital to our success.
"When we identify a player, we always hope that they can settle into things as quickly and as easily as possible, but like so many other players during the season, CAMERON did a great job from the very beginning."
He is a popular guy and we are happy that he sees his future with us.
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A year ago Celtic looked to the relatively unknown Ange Postecoglou to revitalise a club in turmoil and he silenced doubters by winning the Scottish Premiership title and League Cup.
After failing to win his first three matches at Parkhead, Postecoglou was knocked out of the European competition in the group stage.
The Australian insisted that he wouldn't change his philosophy of attacking, entertaining and relentless football and that he rebuilt a struggling side who were destroyed by Rangers the previous season.
The title win came at Tannadice on a Wednesday night in May, with the manager promising this was only the beginning for him.
Click here to read what the Celtic boss has to say about his first season in charge.
The transfer window will open on June 10 and close on September 1 in Scotland.
Sky Sports has a dedicated Transfer Centre website where you can keep up-to-date with all the latest news. Sky Sports News gives you the ins, outs and analysis of sports.
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