We analyze the rotational demagnetization of artificial spin ice, a recently realized array of nanoscale single-domain ferromagnetic islands. Demagnetization does not anneal this model system into its antiferromagnetic ground state: the moments have a static disordered configuration similar to the frozen state of the spin ice materials. We demonstrate that this athermal system has an effective extensive degeneracy and we introduce a formalism that can predict the populations of local states in this icelike system with no adjustable parameters.