How to install stem?
 

At first I thought this would be trivial, but now I wonder if I'm missing a part or doing it wrong.

At the base of the stem, there's a little weight that the allen bolt threads through. I figure this weight must attach or "catch" onto something in the headset/headtube. When I put the stem into the headset, there is nothing for the weight to engage on, and I can't screw in the top allen bolt. Am I missing something?

It isn't a weight. It is a wedge. When you tighten the stem bolt, it will move the wedge up. Since it is cut at an angle, it moves up and "out" slightly, wedging itself against the inside of the steerer. The trick is to have the bolt just loose enough to be able to fit the stem into the steerer, but tight enough that the wedge wont just spin witht the bolt as you tighten it. Don't forget to coat the stem, wedge and bolt threads with a thin layer of grease before inserting it.

On other stems I've seen, it is wedge shaped, but this one looks different than others. It's like a cyclinder with a smaller circle for a top and a larger circle for the base (rolo candy shaped). When I screw it tightly on with the stem out of the bike, it wedges itself into the stem's bottom opening. I don't know if this helps, but I think the bars and stem came off a Specialized. I'll post a pic if my explanation is unclear.

You need help
 

I don't want to seem nasty, and I really don't intend this as a flame. But if you don't know how this works you need to take your bike to a shop to have the stem installed. This is for your own safety. Do it wrong and the bars can come off the bike, or twist in the steerer tube, and you can't steer, big trouble.
Please go to a shop to have the work done.
Many shops also offer a bike mechanics class in the winter, find one and sign up, then do the work your self.

Same principle as explained above.
Rolo candy is a "WEDGE".
As the screw tightens, the wedge is driven into the I.D. of the stem thereby expanding the stem in the steer tube.
Second, and more importantly, get a new stem.
They stopped making that style in the seventies for a reason.

Thanks, worked out great. I can see why the new style is easier to pull out and put in. It's a very pretty stem and it'd be a waste not to use it.

leob1: I'd love to take a class on mechanics to learn tips and tricks, but I feel bikes are simple enough to figure out with some paitience and common sense. No nastiness or offense taken :)