Recently, significant interest has emerged in fabricated systems that mimic the behavior of geometrically-frustrated materials. We present the full realization of such an artificial spin ice system on a two-dimensional kagome lattice and demonstrate rigid adherence to the local ice rule by directly counting individual pseudo-spins. The resulting spin configurations show not only local ice rules and long-range disorder, but also correlations consistent with spin ice Monte Carlo calculations. Our results suggest that dipolar corrections are significant in this system, as in pyrochlore spin ice, and they open a door to further studies of frustration in general.