Anthony, who initially joined the club on loan last season prior to it being curtailed, has previously been a full-time scholar at Fleetwood Town before joining Celtic ahead of the 2020/21 season.
He said: "I'm delighted to be here. I loved the place last year and the fans are fantastic. I was delighted when the gaffer called me and can't wait to get started again. The lads are fantastic and the club has real ambition."
Manager Nathan Rooney added: “I’m delighted Willo has joined us permanently. He was fantastic for the group when he came in last season. The fans already know all about his energy, but he possesses real quality too and can adapt to several positions which is important to me."
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