A standard solution from the Internet is to fill the adapter with epoxy, and that's what I did. But I did one better. I also sunk some scrap metal into the epoxy: some screws from my scrap screw collection as well as leftover bits of a 3/8" steel rod. This increased the solidity of the result while decreasing the amount of epoxy that would need to be used, and hence decreasing the cost.
The epoxy I used was JB Weld, which is nice and heavy. I recommend a slow-curing epoxy, not the five-minute kind, so you don't have to work quickly.
The first photo shows the strengthened adapter. The second shows the "before".
Step 1: Find Bits of Metal and Mix the Epoxy
Then, mix up the epoxy for one side of the object (if the object has more than one side). Try to keep the proportions as per directions--1:1 in the case of JB Weld--and make sure that everything is thoroughly mixed. In the case of an epoxy where the two parts are different colors, like JB Weld, this is very easy to tell. Keep things well ventilated. I like to mix up epoxy on a paper plate.
Step 2: Put in the Metal and Epoxy
Pop any bubbles and smooth a little. If the epoxy is slow curing, you don't need to smooth it very much--gravity will do it for you. Wipe off any excess epoxy.
Step 3: Leave to Cure, and Repeat
You may want to monitor from time to time to make sure things aren't lopsided.
Step 4: Repeat If Necessary
The photo shows the first side. The second side is curing more slowly--I think I didn't get the proportions quite right. JB Weld turns duller in color once it's cured. The photo shows the side that's done.
The resulting adapter should be much, much more stable. (I haven't tried this one yet, but I have used another one I did earlier.) It also has an impressive "expensive-feeling" heft, since it now weighs in at 4.5 oz.