is it a fungus on my rose?

This sounds interesting.
Well if it was mildew, getting rid of it by deadheading your roses is a surefire technique.--the rose heads are something you would normally remove anyway, for the plants benefit.
Boy, wouldn't it have been nice to have seen a picture, though.

I find it quite a coincidence that the "mildew" was only on the stems (did I read that right?). I didn't read that the "mildew" was on any leaves. Personally I find this odd. The mildew attacks that I have seen in my own garden have begun on the leaves and spread from one to another, sometimes involving the stems very slightly.

The distribution (on the stems near the end where new growth and flowers are) reminds me of one thing immediately.--It is the textbook location of aphids, and there is a sort of aphid that resembles middew, that is to saw, white and powdery in appearance. It makes me wonder if it could have been that.
I had an infestation of this on a plum tree this past summer for the first time ever.

I suppose that it is academic now that you have removed the ends of your rose stems; however, to be sure I would re-check every stem, bud and leaf just to be sure all is gone. If there is any more, please take fotos.

I am sorry that I can't recall the name of that aphid sort at the moment.

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Sjoerd http://www.volkstuindersvereniginghoornenomstreken.nl/Page11.html

It is absolutely mildew on the leaves.
There are some commercially produced sprays especially for this disease; however, I would remove and burn the affected bits.
I personally feel that one of the most important aspects of treating this disease is the thorough cleaning of the ground at the base of the rose, picking up every dead leaf, every petal ...in short all debris from your infected rose that's lying on the surface of the ground. This is so that the plant (and other plants in the area for that matter) won't become (re-)infected.

Another thing...if I see correctly, it looks as though that lovely rose is situated in a corner... is that right? If so it could possible be a key reason that your rose has become infected.
What I mean by this is that when in a corner, there is restricted air flow because of the two solid walls on two of the four sides. Roses do need good air flow artound the branches and leaves...it's one of the reasons that you prune.

You need remove that head and leaves as well as spray that little gem as soon as you can.

Just one more thought, why don't you ruffle carefully inspect the mulch after cleaning it's surface to to see if there is some of that mildew undser the upper layer. I once discovered a mildew source IN the mulch under one of my rose plants and had to remove the entire layer and replace it with new.

You know, I now have three roses. One growing up against my little garden shed and the other two out in the elements, and the one that gets most bothered by pests of all sorts from mildew to aphids is the one next to the house...every time.
Well, I've had enough and I am going to set it out in the open this year. I will place it at the end of the clematis rack and see if it fares any better there.
Good luck with your rose...do not delay any longer in taking steps to correct this problem.

NOTE: If you want to have some fun (and proove that it is nildew), there is something that you can do...if you have a simple microscope, like the ones folks buy for their children.

--Simply scrape a litttle bit of the white stuff off the stems and leaves, put it on a slide, add a drop of calcium hyfroxide...wait about 20 minutes then go and have a look. The mycelial elements will be easily visible if it is indeed a fungus. They will appear elongated with a barely perceptable greenish tinge to the edges of the "branches", as I recall.
T
his may sound a bit clinical, but it is definative proof and something interesting for your kids...and perhaps you as well.

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Sjoerd http://www.volkstuindersvereniginghoornenomstreken.nl/Page11.html