We report on the realization of a matter-wave interferometer based on single-photon interaction on the ultranarrow optical clock transition of strontium atoms. We experimentally demonstrate its operation as a gravimeter and as a gravity gradiometer. No reduction of interferometric contrast was observed for a total interferometer time up to ∼10 ms, limited by geometric constraints of the apparatus. Single-photon interferometers represent a new class of high-precision sensors that could be used for the detection of gravitational waves in so far unexplored frequency ranges and to enlighten the boundary between quantum mechanics and general relativity.