Ok, I've done some of the things you kind folks suggested – all to no avail, I'm afraid:
1. I tweaked the level of the float. My toilet probably uses less water than before, but it didn't solve the problem.
2. I cleaned the flapper and the edge of the ring beneath the flapper. Nada. Still recycling water like it was before.
3. I replaced the flapper with a new one. Didn't help. Still recycling water every few hours.
Is it time to replace the whole toilet assembly, or is there something else I can try first?