Handbag construction woes...

[atygir.livejournal.com]

I want to make a small, circular handbag for my sister for Christmas. I've got a simple design sketched out (it's basically two circles connected by a wide strip). It won't be getting any heavy-duty use, but I'm worried that whatever I make it out of it's going to be too flimsy, and will collapse in on itself as soon as you put anything in it. I was thinking of sewing cardboard or something into the base and lining to give it a bit more structure. I've never done anything like this before, and have no idea of how I would attach it, or if it would even work. Any thoughts?