I would say no not into the old wood. You could try cutting right back to just before the old wood? But I dont think it will stop it? as this old rose is prone to the disease and its in the soil and debris around it so even after hard prunning I would expect it to come back?

I don't know what you've been treating the rose with, but go and buy some Roseclear 3 (not any other Roseclear, the 3's important) and a sprayer. Sounds as if your rose has mildew as well as black spot - mix the roseclear to fungicidal strength and spray every 2 weeks till run off, under the leaves and the stems too. Pruning designed to open out the rose would be good too, to enable more air circulation around its stems and leaves, but you don't say which rose it is, so can't advise further on pruning.