It doesn't have thorns does it? I don't think it's a rose, mostly because the way it is putting out such skinny little cross branches.
I think the first thing to do would be to just remove the dead stuff. Cut it down to living cane/stems. (green in the middle) and then water it properly. Most plants like the top inch of soil to be dry before rewatering.

I recommend reposting this on the Name Taht Plant Forum and add a picture of a green leaf.

This post was edited by lola-lemon on Thu, Jun 6, 13 at 22:50

Thanks, lola-lemon! When you say cut it down, do you mean remove all the foliage entirely? The remaining three green leaves are at the tip of an entirely brown branch covered in dead leaves.
I'll definitely post a pic on NTP when the sun comes up tomorrow!

Oh! I didn't realize the 3 green leaves were on the top. Just strip the brown leaves and keep it watered and see what it is --- then decide how to manage it. Some bushes allow renovating, some don't.
It might be a gardenia and they can be hard to keep alive in certain climates. That's just a guess made looking at dead leaves tho... so go ahead and post the green leaves pictures over on the name that plant forum.
Good luck!

It's not a rose. Looks like some kind of shrub. There's one called Coppertina that has leaves that color. Or maybe some type of viburnum.