The heaviest elements in the periodic table are synthesized through the r-process, but the astrophysical site for r-process nucleosynthesis is still unknown. The major current candidates are ordinary core-collapse supernovae and neutron star merger. Ancient, metal-poor ultra-faint dwarf galaxies contain a simple fossil record of early chemical enrichment that provides the means to study clean signatures of nucleosynthesis events, and thus, can yield unique information on the origin of these processes. Previously, extremely low levels of neutron-capture elements were found in the metal-poor stars in ultra-faint dwarf galaxies which supported supernovae as the r-process site. Based on Magellan/MIKE high-resolution spectroscopy, we have determined chemical abundances of nine stars in the recently discovered ultra-faint dwarf Reticulum II. Seven stars display extremely enhanced r-process abundances, comparable only to the most extreme r-process enhanced metal-poor stars found in the Milky Way's halo. The enhancement is also 2-3 orders of magnitude higher than that of stars in any of the other ultra-faint dwarfs. This implies the neutron-capture r-process material in Reticulum II was synthesized in a single prolific event that is incompatible with r-process yields from ordinary core-collapse supernovae but consistent with that of a neutron star merger. This would be the first signature of a neutron star merger in the early universe which holds the key to finally identifying the r-process production site. Furthermore, such a single r-process event is a uniquely stringent constraint on the metal mixing and star formation history of this ultra-faint dwarf galaxy.