The 23-year-old midfielder Wedderburn, who left Central Park last season, becomes the Scottish Cup holders’ first signing of the summer.
Manager John Hughes told the club’s website: “He is a big strong lad with good speed and agility. Nat will be a great addition for the club.
“He is joining a great club at a fantastic time and we look forward to him developing in to a regular first-team player. He is stepping up a few levels, but we know he is well capable of doing so.”
Wedderburn added: “I don’t expect to go straight into the team. It’s like when you go to any new club, you have to earn that right, but I am here to work hard and play my part.”
Meanwhile, it has been revealed that the club have been unable to secure the services of striker Ofere for next season after talks over a new deal broke down.
Ofere joined Caley Thistle on a short-term deal in February and scored on his debut against Motherwell. He went on to score four more times, including in the 3-2 win over Celtic in the Scottish Cup semi-final.