We present a search for new members of the 300 km s-1 stream (300S) near the dwarf galaxy Segue 1 using wide-field survey data. We identify 11 previously unknown bright stream members in the APOGEE-2 and SEGUE-1 and 2 spectroscopic surveys. Based on the spatial distribution of the high-velocity stars, we confirm for the first time that this kinematic structure is associated with a 24º-long stream seen in SDSS and Pan-STARRS imaging data. The 300S stars display a metallicity range of -2.17 < [Fe/H] < -1.24, with an intrinsic dispersion of 0.21-+-0.09-+0.12 dex. They also have chemical abundance patterns similar to those of Local Group dwarf galaxies, as well as that of the Milky Way halo. Using the open-source code galpy to model the orbit of the stream, we find that the progenitor of the stream passed perigalacticon about 70 Myr ago, with a closest approach to the Galactic Center of about 4.1 kpc. Using Pan-STARRS DR1 data, we obtain an integrated stream luminosity of 4x10^3 L_sun. We conclude that the progenitor of the stream was a dwarf galaxy that is probably similar to the satellites that were accreted to build the present-day Milky Way halo.