The Irishman agreed a three-year contract with the Ayrshire club, having left Sky Bet League 1 side Crawley Town at the end of last season.
Connolly has played in Scotland previously - he was on loan at St Johnstone in 2010 - but the defender is more notable for having been signed for £1m as a teenager, when Bolton Wanderers took him from Wolves.
Now 22, Connolly turned down offers to remain in England and instead agreed a move to Rugby Park. "I had four or five offers down in England but this felt right," he said. "If we are away on a windy Tuesday night or at home on a sunny day, I can get the ball down and play. The manager wants to play and that's a big thing for me."