Most digital cameras operate pretty well below freezing. After all, unlike old mechanical cameras that were full of moving parts and needed to be winterized" with special oils and other tricks, todays digitals are almost entirely electronic. There are virtually no moving parts, aside from zoom and shutter button, so less to freeze up.
So, the problem is not with the camera. Its with the battery. In cold weather the batterys little electrons start to move every more slowly, eventually grinding to a complete halt if they get cold enough. And when that happens the camera ceases to operate.
That said, Ive used pretty modern cameras (film-based, but with plenty of electronics) down to minus 20 F or so without many problems. The trick, and youre already onto this, is simply to keep the camera as warm as you can. For a compact digital camera, that isnt much of a problem. Theres bound to be a pocket that can hold it, or you can hang it next to your chest in a case with a neck strap and zip a jacket over it. Leave the camera out in the open for 40 to 50 minutes, and, depending on the temperature, it will shut down.
The other thing to do is carry spare batteries and keep the batteries warm. After all, thats really the objective. Keeping batteries in a pocket will keep them at pretty close to body temperature, and theyll work just fine. Then, swap the batteries out of the camera every hour or so. This is easier and more affordable with a camera that uses batteries such as AA size, versus one that uses a proprietary (and, expensive) battery. But for any camera its always possible to buy an extra battery or three.