Tom suggested it could be spider mites.
Bingo! I had seen some webs on the plant but had assumed they were from beneficial spiders. When I looked more closely, sure enough, there were spider mites on the undersides of the leaves. Apparently they were sucking the plant dry.
It was too late that night to take chemical measures, so I blasted the plant with the hose, which is often a good first line of defense against insects. Then later I treated the plant with a pyrethrin-based insecticide. I would have preferred an oil-soap spray, but I happened to have the other product handy.
I'll reinspect in a few days and see if the mites have disappeared. If not, I'll need to treat them again.