The pyrochlore material Ho2Ti2O7 has been suggested to show "spin ice" behavior. We present neutron scattering and specific heat results that establish unambiguously that Ho2Ti2O7 exhibits spin ice correlations at low temperature. Diffuse magnetic neutron scattering is quite well described by a nearest neighbor spin ice model and very accurately described by a dipolar spin ice model. The heat capacity is well accounted for by the sum of a dipolar spin ice contribution and an expected nuclear spin contribution, known to exist in other Ho3+ salts. These results settle the question of the nature of the low temperature spin correlations in Ho2Ti2O7 for which contradictory claims have been made.