I would like to mount a small point and shoot camera on my handlebars, so I can take some short videos while riding. Has anybody tried to do this?

I know about the Hero camera, but would like to use what I already have.
I've been to photo store, and they have clamps specifically for that reason, but they are really big and heavy, and not suitable to take touring.

I tried putting the camera on a small tripod and zip-tying it to the bars, but it doesn't seem very stable. Any ideas?