Less than two ounces is probably dreaming.
This Panasonic camera shoots decent photos and video, is waterproof, shockproof, and freezeproof, and costs about $150:
It weighs 154 grams without the battery. Just turn it on the "intelligent auto" mode and fire away.
You can spend the extra money on a spare battery (if your trips go more than a day or two), and a 16GB SDHC card.