How do you shoot in freezing fog...?

This is a problem I've been facing lately, and one I don't think I've encountered until the past month.  Hopefully someone else has dealt with this and found a solution.

It's winter in the PNW, and we've had a lot of temperature inversions lately.  At low altitudes, it's been cold and dreary, with low fog cover;  in the mountains it's been bright and warm, without a cloud in the sky.  I've been taking advantage of the situation, snowshoeing in a tee-shirt during the day, and stargazing with my camera at night.  (With no clouds to hold the warmth in, it radiates away and gets very cold at night.)

What happens is I get a thin layer of ice forming on the tripod legs, the camera body, or at least parts of it, and on the front of the UV filter.  The last part fogs the image.

I can wipe the lens down and get rid of it, but wait five minutes and it's back!

Longish exposures, like star trails, haven't really been possible because of this.  Sometimes the fog wants to freeze so quickly that I'll have to wipe the lens down vigorously between exposures, which means multiple images won't stack without being aligned.

Has anybody else come up with a solution for freezing fog?