Maryanne,
If you do nothing, you will lose the plant. Rot coming down from the top and up from the bottom will eventually kill it. I would cut off the tip as far down as the rot, and continue down until you see clear, white wood. Then, I would cut off the bottom section at the top of the rot coming up, and again cut the good section you have removed until you have clear, white wood. Hopefully, you will have a clean section that is long enough (about 18cm or more) to root as a center cut. Treat the cut ends with a fungicide and allow the cut section to callus over for about a week or so, and then pot it in fast draining mix with the old leaf scars looking like a smile. Water it once and put it on a heat mat under grow lighting and see if it will root for you. If you don't have a heat mat/grow lighting, you should keep it indoors in a warm spot near a sunny window. Don't water again until leaves form, and then water sparingly.

It may take a while for leaves to develop. Watch for signs of rot. Good luck.
Mike

Thanks Mike for answering so quickly, but bad news: I've just cut it and the rot runs vertically all along on one side and in the middle of some bits. I'm going to try anyway but it doesn't look good. I wish I had dared to do it before. :( I'm hanging on to hope for the 2 little sprouts, though. Do you think they could take in sand? I have room inside for a very small pot. Do you think there's any chance for them if I pluck them off the stalk and put them in sandy soil? As I said they're about 2 cm long.

Maryanne,
Honestly, the chances are slim that the two small branches will root. It is hard to root green cuttings (cuttings with green bark). It is easier to root more mature cuttings. BUT, you have nothing to lose. However, don't use sand to root them. Your chances are better in a mix of potting soil and pumice (or perlite) about 1:1 or 2:1. You can also use coir. Use a rooting aid with a fungicide like Rootone. I would place them about 1 cm. deep and water once well. Then leave them alone in a warm, sunlighted place to see if them will root.

I tried this once with a similar size cutting and actually got it to create a single root, but it did not survive.

Another option we have here in Florida is to send things that size to Florida Colors. Luc will graft the tip onto established root stock. If it takes, you will have saved the plant. You may want to see if someone in your area can graft the tips onto plumeria stock.

As I say, go for it. You have nothing to lose.

Mike

Ok, fingers crossed then!

Thanks again for all the advice, Mike. I'm not on the computer much these days, but I'll try and update, especially if there's any good news. (Can't help holding on to hope!)