Rosa multiflora perhaps? Over time it will probably crowd out the azaleas. If it is multiflora it reproduces by seed and by forming new plants that root from the tips of arching canes that contact the ground. I've included a link so you can see if this is the rose to which you're referring. If it is, this link also contains some control methods, although it can take a lot of persistance. Good luck.

Heather

Here is a link that might be useful: Multiflora rose

The most probable damage to the azaleas would come from them being overgrown and shaded out. Since azaleas will take a certain amount of shade, it's not too likely if you take steps to control the rose.

The method of control for the roses that I would suggest is for you and your DH to wear sturdy, long-sleeved shirts, long pants and gloves, and working together, cut out stems as close to the base as you can, or perhaps more accurately descibing conditions - as far back from an end as you can. If you keep cutting, and are careful for the azaleas, you should eventually find your way back to the where the base of the roses is. If, as suggested, it is a multiflora, then you may have multiple bases of origin. When you get back to the roots, then you can either dig them out or use Round-Up or Brush-B-Gone on the cut ends. Digging is perhaps more sure. Any new shoots you see should be quickly dealt with - you may get sprouts from the roots. I would very carefully pile up the stems as you cut them and burn the pile when it is dry - my experience of rose thorns is that all the thorns don't break down in a compost pile and will "bite" you later on as you handle the compost. If you don't use your current compost pile for a year or more, then never mind the last.

Thanks to both Heather and dibbit. We are going to start cutting it back tomorrow. We'll be sure to put Round-Up on the ends. Didn't know that maybe all the thorns would not decompose. Will certainly watch out for that. Thanks for the info. Right now my hands and arms look like I've been in a cat fight- scratches all over. Will certainly wear protective clothing when dealing with that rose. I can't get rid of it fast enough. THANKS !!

If you get those tubes that florists use for keeping flowers fresh, fill them with undiluted Round up, cut back the stems as far as you can, and then stick the tube on the cut end (still attached to the main plant, of course), the plant will soak up the Round up, and die back even farther than you can reach. It's not perfect, doesn't kill all the way to the ground, when the plant roots in several places, but it's a help. A friend of mine calls this rose "kudzu with thorns." Good luck!