How old is the plant in question? Some canes will start to wood over as they get older, but that could also be canker...

This rose is maybe 4 years old, but has been transplanted twice in that time.

Transplanting shouldn't matter unless you cut all the canes off when you did it. It's the age of the canes that's important. A 4 year old cane could begin to get woody/barky covered. I turned and brightened up the photo to see it better and this looks like bark not canker to me.

Thank you for the replies.

I was hoping that was the case.

It was just that the woody growth looked, to me at least, very different from my other roses and I was worried that if this was diseased, it could spread.