All fixed. The original gasket between the bowl and the top half of the carb was deteriorating, there was a crack around it that showed daylight, and some bits of it got into the little valve that the float actuates, I'm not sure what it's called. Needless to say that the poppyseed-sized bits of rubber kept the valve from closing completely and thereby caused the carb to dump gasoline everywhere.
I cleaned out the float-actuated valve and made a new gasket out of rubber-fiber gasket material. The engine now runs like new, and the carburetor no longer leaks gas. I am now a happy camper once more. What's more, I can now tell people that I fixed a carburetor on my first attempt, and it was the first time that I ever actually disassembled a carb, of any kind, EVER. No missing pieces, either. Yay!