I have had the EXACT SAME PROBLEM.
What I have done is, lessned up on my regular curls, and started to do more like preacher curls (palms down, or faced inwarsd as opposed to palms facing down, etc).
This has helped me, first becuase I need to work on my outer bicep a bit more and get it developed,
Making this change, emphasizing more "palms down or facing in" curls has helped make the pain go away.
Try this. Go for several weeks doing NOTHING BUT palms down and palms inwards versions of various curl exercises. Then start to incorporate the palms up stuff back into the routine. Never go back to ALL pamls-up, which is what most guys do and leads to over-stressing certain parts of the wrist (and developing the arms in such a way that this promotes the pain).
This has just about fixed it for me.