Cureton suffered the injury while scoring on his debut in Saturday's 2-2 draw with Burton and was immediately substituted.
The fullextent of the problem was revealed on Tuesday following a series of scans and the 37-year-old will now begin rehabilitation.
"Jamie will now start a five-six week rehab process," physio Ian Weston told the club's official website.
"The good news was that there was minimal soft tissue damage and he won't require any surgery. This is Jamie's first dislocation so we will be working hard to strengthen the shoulder after we take it out of the sling.
"We'll aim to get him out of the sling by the end of the week and that's when we'll begin to strengthen the shoulder."
- Sports & Recreation