We present a physical model that accounts for the motion of rapidly rotating curling rocks. By rapidly rotating we mean that the rotational speed of the contact annulus of the rock about the centre of mass is large compared with the translational speed of the centre of mass. The principal features of the model are: ( i ) that the kinetic friction induces melting of the ice, with the consequence that there exists a thin film of liquid water lying between the contact annulus of the rock and the ice; ( ii ) that the curling rock drags some of the thin liquid film around the rock as it rotates, with the consequence that the relative velocity between the rock and the thin liquid film is significantly different to the relative velocity between the rock and the underlying solid ice surface. Since it is the former relative velocity which dictates the nature of the motion of the curling rock, our model predicts some interesting differences between the motions of slowly versus rapidly rotating rocks. Of principal note is that our model predicts, and observations confirm, that rapidly rotating curling rocks stop moving translationally well before rotational motion ceases. This is in sharp contrast to the usual case of slow rotation, where both rotational and translational motion cease at the same instant. We have verified this and other predictions of our model by careful comparison with the motion of actual curling rocks.