A slingshot, as has already been mentioned, is a very efficient method of accelerating shot, or pebbles if you must, to hunt small game.
A Sling is something that hasn't been mentioned. It takes an awful lot of practice to get good enough to hunt reliably with one, but it can be done. It is more forgiving that a slingshot in terms of what ammo you use, so pebbles will work just as well as shot for this. It is lighter and smaller than a slingshot.
A bolas requires no ammo at all, and is nothing more than rope in terms of carrying capacity. your range is reduced, but takes less practice than a sling to get accurate with. Most techniques of using these will tangle, rather than kill, your prey, so you will still need to kill it by hand (you have a knife, I assume).
Roping can work on slightly bigger prey, and some birds, using a rope or net to tangle your prey. A small-celled net can work on very small game. Again, your range is reduced even further here, but requires much less practice.
Thrown implements, ranging from a rock to a spear can also work, with varying range and skill requirements. A spear can have a decent range, and can be accurate enough to hunt with well, and should be re-usable with minimal maintenance. If the shaft breaks though, you will be without a hunting tool.
Of course, the legality of all these methods is dependent on where you are hunting, these recommendations are from a purely practical viewpoint. I would still go with a bow out of choice, myself.