The heaviest chemical elements in the periodic table are synthesized through the rapid neutron-capture (r-) process but the astrophysical site where r-process nucleosynthesis occurs is still unknown. The best candidate sites are ordinary core-collapse supernovae and mergers of binary neutron stars. Through their stars, 13 billion year old ultra-faint dwarf galaxies preserve a "fossil" record of early chemical enrichment that provides the means to isolate and study clean signatures of individual nucleosynthesis events. Until now, ultra-faint dwarf galaxy stars displayed extremely low abundances of heavy elements (e.g. Sr, Ba). This supported supernovae as the main r-process site. But based on new spectroscopic data from the Magellan Telescope, we have found seven stars in the recently discovered ultra-faint dwarf Reticulum II that show extreme r-process overabundances, comparable only to the most extreme ancient r-process enhanced stars of the Milky Way's halo. This r-process enhancement implies that the r-process material in Reticulum II was synthesized in a single prolific event. Our results are clearly incompatible with r-process yields from an ordinary core-collapse supernova but instead consistent with that of a neutron star merger. This first signature of a neutron star merger in the early universe holds the key to finally, after 60 years, identifying the cosmic r-process production site, in addition to being a uniquely stringent constraint on the metal mixing and star formation history of this galaxy from the early universe.