Chemically peculiar stars in dwarf galaxies provide a window for exploring the birth environment of stars with varying chemical enrichment. We present a chemical abundance analysis of the brightest star in the newly discovered ultra-faint dwarf galaxy candidate TucanaIII. Because it is particularly bright for a star in an ultra-faint Milky Way (MW) satellite, we are able to measure the abundance of 28 elements, including 13 neutron-capture species. This star, DESJ235532.66-593114.9 (DES J235532), shows a mild enhancement in neutron-capture elements associated with the r-process and can be classified as an r-I star. DES J235532 is the first r-I star to be discovered in an ultra-faint satellite, and Tuc III is the second extremely low-luminosity system found to contain r-process enriched material, after Reticulum II. Comparison of the abundance pattern of DES J235532 with r-I and r-II stars found in other dwarf galaxies and in the MW halo suggests a common astrophysical origin for the neutron-capture elements seen in all r-process enhanced stars. We explore both internal and external scenarios for the r-process enrichment of Tuc III and show that with abundance patterns for additional stars, it should be possible to distinguish between them.