I've replaced the 555, rewired the circuit, and tested the resistance of the capacitor; the result is the same: the green LED is always powered on, the red is always powered off. I don't know what's meant to happen when I tested the resistance of the capacitor, but it started at about 2M and its resistance rose until it was off the scale of the meter (over 50M). I'm guessing that's a result of the multimeter charging the capacitor.
I'm not sure what's wrong with the circuit, but I have followed the circuit diagram (except for the fact that I'm using a ceramic capacitor rather than an electrolytic one) and I'm using brand new components, and have tested most of them. Are you sure there isn't still a problem with my circuit?