If it is dead I would cut it. However, some types of roses, and older roses, develop woody canes that are not dead. So not really sure not knowing your rose.

Unfortunately, I don't know what rose it is. The rose is a very dark red, almost black, and it's very thorny.

Well it looks like the new cane is coming up from below the graft so it may nebe root stock, maybe just wait and see.

It looks like you have cane borer damage (the holes in the dead wood). You should cut that down, it looks dead enough that it will probably crack apart. You can wait and see if the new growth is good or not. Maybe it will grow more, but if it only shoots out one little cane this season it may be a pretty sad looking rose. If it's in a prominent spot you may want to look at replacing. If it's in a spot that you can wait and see, that is always an option. But sometimes in can take a few seasons to rebuild some substance to it.

Cane borer damage: I thought that was just a decorative detail! Does this spread? I’ll do some research - thank you. It’s behind a knockout rose and the flowers it gave me were prickly but so beautiful that I will keep it and see what happens.