So I did a bit of blasting with a hairdryer, fiddling with pipes etc., and then went outside and tipped a kettle of boiling water over the pipe. It make alarming crackling noises, which was probably the ice melting, and then the sink began to drain freely. Phew.
Thanks to Wesley and 1690 an' all thon for the advice.
So now I have to try to stop it from icing up again. Here are my plans:
1. Stop the dripping tap. Now, I can't actually stop it dripping, but I figure if I catch the drips in the washing up bowl and then tip them all down at once, that'll be ok - it's just when they're little drips they freeze more easily.
2. Construct some kind of lagging device to keep the outside pipe warm.
You see, what I'm worried about is if I go home for Christmas and then it ices up and the washing basin overflows all over the floor. I'm trying to think up some kind of device which would catch the drips and then periodically tip them all out into the sink, but I can't.