We present far-ultraviolet photometry obtained with the Hubble Space Telescope of the low-mass X-ray binary 4U 0513-40 in the globular cluster NGC 1851. Our observations reveal a clear, roughly sinusoidal periodic signal with $P ≃ 17$ min and amplitude 3%-10%. The signal appears fully coherent and can be modelled as a simple reprocessing effect associated with the changing projected area presented by the irradiated face of a white dwarf donor star in the system. All of these properties suggest that the signal we have detected is orbital in nature, thus confirming 4U 0513-40 as an ultracompact X-ray binary (UCXB). All four confirmed UCXBs in globular clusters have orbital periods below 30 minutes, whereas almost all UCXBs in the Galactic field have orbital periods longer than this. This suggests that the dynamical formation processes dominate UCXB production in clusters, producing a different orbital period distribution than observed among field UCXBs. Based on the likely system parameters, we show that 4U 0513-40 should be a strong gravitational wave source and may be detectable by LISA over the course of a multi-year mission. ; Comment: 17 pages, 7 figures, Accepted by ApJ