Single-photon photoemission of electrons for incident photon energies below the surface work function is reported for a single-crystal Cu(111) photocathode. Spectral characterization of the quantum efficiency and mean transverse energy of the emitted electrons is shown to be consistent with emission from a thermalized hot electron distribution photoexcited on sub-picosecond time scales into an upper conduction band at the L-point of the Brillouin zone of copper. To our knowledge, this is the first time that such excited-state thermionic emission has been observed from a planar metal photocathode, and then from a commonly used photocathode material. The results, therefore, indicate the potential importance of such band structure dependent sub-threshold photoemission mechanisms on the performance of photocathodes employed for the generation of short electron pulses using sub-picosecond laser pulses. Consequently, the combined experimental and theoretical work presented in this paper contributes to solid-state photocathode-based research aimed at our understanding and selection (or discovery) of high brightness photo-electron sources required for many electron-based diffraction, imaging, and accelerator applications.