We calculate properties of neutron star matter at subnuclear densities using an improved nuclear Hamiltonian. Nuclei disappear and the matter becomes uniform at a density of about 0.6n(s), where n(s) of about 0.16/cu fm is the saturation density of nuclear matter. As a consequence, the mass of matter in the crusts of neutron stars is only about half as large as previously estimated. In about half of that crustal mass, nuclear matter occurs in shapes very different from the roughly spherical nuclei familiar at lower densities. The thinner crust and the unusual nuclear shape have important consequences for theories of the rotational and thermal evolution of neutron stars, especialy theories of glitches.