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Superconducting nanowires used in single-photon detectors have been realized on amorphous or poly-crystalline films. Here, we report the first use of single-crystalline NbN thin films for superconducting nanowire single-photon detectors (SNSPDs). Grown by molecular beam epitaxy (MBE) at high temperature on nearly lattice-matched AlN-on-sapphire substrates, the NbN films exhibit a high degree of uniformity and homogeneity. Even with relatively thick films, the fabricated nanowire detectors show saturated internal efficiency at near-IR wavelengths, demonstrating the potential of MBE-grown NbN for realizing large arrays of on-chip SNSPDs and their integration with AlN-based χ (2) quantum photonic circuits.