  The application of a magnetic field along the [111] direction in the spin ice
compounds leads to two magnetization plateaux, in the first of which the ground
state entropy is reduced but still remains extensive. We observe that under
reasonable assumptions, the remaining degrees of freedom in the low field
plateau live on decoupled kagome planes, and can be mapped to hard core dimers
on a honeycomb lattice. The resulting two dimensional state is critical, and we
have obtained its residual entropy -- in good agreement with a recent
experiments -- the equal time spin correlations as well as a theory for the
dynamical spin correlations. Small tilts of the field are predicted to lead a
vanishing of the entropy and the termination of the critical phase by a
Kasteleyn transition characterized by highly anisotropic scaling. We discuss
the thermally excited defects that terminate the plateau either end, among them
an exotic string defect which restores three dimensionality.
