The 28-year-old was recently offered a new contract by Binos boss Jocky Scott but has rejected the deal to sign for Morton.
Moore told the Tele: "I met with Ross on Friday night and he has signed for us for next season.
"He was very keen to come here, and we've agreed a one-year deal. I'm delighted to get him as he was a player I was looking to sign at the beginning of last season.
"He can play in a number of positions, which is a bonus with us going with a smaller squad next season.
"But I see him as a left-back and that's where we'll be looking to play him.
"When I had him at Stirling he was the top left-back in the Second Division, and he was one of the best in the First Division this season."