Your rose has been invaded by a mossy rose gall wasp. His (or her) visit resulted in damage to the stem; a biochemical reaction occurs between the insect and the plant which results in the formation of this strange tissue growth. The wasp (diplolepis rosae) is common in late spring, when lush soft growth is appearing on the roses.  Technically, this is a harmless growth, but it is unsightly and certainly weakens the stem on which it is located. Clip off the stem well below the growth (best at a five-leaf junction) and discard, do not compost, the piece. No sprays are necessary or effective. For an interesting look at the various other rose gall wasps and a good close-up of your particular invader, check out bugguide.net. Search on Genus Diplolepis. Happy rose growing! 

Thank you so much for the plethora of information on this growth. Haven't seen any wasps in my yard, but will keep an eye out. Will also check out the site you suggested. Again, many thanks!

We're talking tiny insects here--you probably wouldn't see the culprit. One thing I didn't mention was that the purpose of the wasp's visit was to deposit eggs in the plant tissue--that larvae would be found inside if you were to dissect the growth. Be sure to discard that infested stem!