I thought it was a Jasmine but the leaves don't look right.

A clematis?

One of the more invasive Clematis would be my guess. I have a similar species and it seeds everywhere.

I'd go for clematis - how big is it and will you be able to disentangle it from the hedge?
Are these current photos or from earlier in the year? my first thought is Clem ternifolia Sweet Autumn. (See http://www.heritageflowerfarm.com/buyPerennialPlants.asp?id=5 about six plants down the list)
One option might be to take cuttings and get them going before you try to transplant - you'll probably need to prune it back anyway to be able to move it.
As far as transplanting goes, wait until it's finished flowering, prune back to release it from the hedge and take as big a rootball as possible.

Thanks for your comments. I took these pics the same day I posted. I can't tell how big it is because it's all twisted throughout the hedge which about 6 feet tall. It's just coming into flower now, not yet peaked.

Clematis microphyllus most likely...

thanks everyone