We have observed, in metamaterial with hyperbolic dispersion (an array of silver nanowires in alumina membrane), a sixfold reduction of the emission lifetime of dye deposited onto the metamaterial's surface. This serves as evidence of an anomalously high density of photonic states in hyperbolic metamaterials, demonstrates the feasibility of an earlier-predicted single-photon gun, and paves the road for the use of metamaterials in quantum optics.