I posted a question about this on Dave's Garden and got some really good responses. The type of rock that I can get from a farm field around here is granite. Two people told me that Granite is hard, but would carve well. You need an angle grinder with a diamond blade. You use the grinder to cut grooves in the rock in a cross hatch pattern, like a giant tic-tac-toe board with lots of squares. Use a chisel to knock out the squares between the grooves. then repeat the process in smaller circles so that the birdbath is deeper in the center than at the edges. At this point, the bottom will have to be smoothed out. I was told that there's a very expensive grinder blade designed for this purpose, but that I could probably use the side of the regular blade to do the same thing. The regular blade isn't designed for this, so it will ruin it quickly and may be dangerous, so this is probably a bad idea. Make sure you wear the proper safety equipment.

Since I haven't done any of this myself, I'm not sure how good this advice is. Maybe someone with more experience can speak up here.