I searched for a long time for an ultralight tool that would work for hunting small game in the backcountry. In the end I decided to go with a slingshot, as it is the lightest effective tool for hunting (next to slings). Slingshots are not for everyone, however, as they require the most amount of skill in order to be effective for hunting. It takes a lot of practice to master a slingshot, and you should only hunt with one if you are willing to put in the time to hone your skills. One thing to note is, slingshots require the highest level of stalking, as you have to get fairly close to prey in order to get a good shot. I would not take a shot at an animal that is more than 20-25 ft away with my slingshot, as there is a real risk of injuring, rather than killing, the animal at those distances. Hunting with a slingshot always runs the risk of injuring an animal. That risk is a lot higher if you are not skilled with the slingshot, so again I stress, develop your skills with target practice before attempting to hunt with a slingshot. When you do hunt with a slingshot, I recommend carrying a sharp knife at all times, so that you may quickly put the animal out of its misery if you happen to stun/injure it. Headshots are key to killing the animal instantly, and with small game, the head is not a very big target. With enough practice, however, one can consistently achieve headshots when hunting.
The slingshots that I found work best for hunting are the traditional slingshots, NOT the modern wrist braced slingshots. Traditional slingshots are more intuitive and allow for much faster shots, especially for moving targets. Personally, I use the traditional chinese style slingshots. These can be acquired from www.dankung.com. These type of slingshots requires you to flick your wrist toward your target at the moment of release. This type of shooting is very intuitive. These are very small slingshots and it is hard to master this style of shooting. However, if you take the time to develop your skills with this style slingshot, it is by far the most effective slingshot for hunting, IMO. They have a steel version that weighs around 4 oz and a Titanium version that weighs around 3 oz. Personally, I prefer the lighter Titanium version. For dankung slingshots, 8 strand bands work better than 4 strand bands for hunting purposes, as they provide more power. Also, you should use 3/8" steel balls for hunting. These can be acquired from Trumark at www.slingshots.com. Lead balls supposedly achieve a higher velocity than steel balls, but I have never used lead balls as they are hard to find.
Slings are even lighter, and with enough practice, are just as effective as slingshots. However, they are even harder to master than slingshots and it takes a LOT of practice to be accurate with them.