Worst case..... pick up a cheap rebel at the camera store, shoot with it for the rest of the trip, and sell it when you get back.
I didn't win the lottery to buy cameras like that mate.. was already a stretch to get this one.
went to a shop tried a different battery and nothing changed. needs to go back.
gonna have to shoot the rest of the trip with my iphone.. 
the camera was out whole day at 1 / 2ºC i guess at night went down to -2. I only had about 600 shots with the camera.. i'm disappointed with canon! Also earlier had a very few drops from light rain. my old 350d had been in worse conditions and survived very well!
I can't afford to go out buying cameras on a whim either.
I suggested buying and selling upon return as a last-ditch worse-case effort and am hoping that something better will help you out.
As to the temperatures, I just can't see mild temperatures like around zero killing the camera. I have mine out for 12 hour stretches at -10C, and several multi-day trips per year outside at those temperatures. The only problems I have had is that after a while the screen gets real slow. The shoulder display stops changing, and then I give up and put the camera away.
What about humidity? Have you tried putting the camera somewhere warm and dry, take the battery out, the lens off, and open up all the doors and flaps, and let it sit for a day? The 6D isn't weather sealed and humidity could be the problem.
BTW, don't let rip with a hair dryer... it is too hot and will cause damage.
Hope this helps....
