I just can't bring myself to believe that the Lithium Ion battery is the problem with my light. Certainly the cold will shorten the battery life, but would you expect a two-hour battery life to be reduced to ten minutes?
My LED blinkers also have LiOn batteries, and they go for about 45 minutes in the very worst cold, or just less than half the expected life. It just doesn't add up; some other part of the Cateye light probably isn't very well-made.
Nevertheless, I think I'll go for a dynamo light setup and fix the problem once and for all.