A search for high-energy neutrinos interacting within the IceCube detector between 2010 and 2012 provided the first evidence for a high-energy neutrino flux of extraterrestrial origin. Results from an analysis using the same methods with a third year (2012-2013) of data from the complete IceCube detector are consistent with the previously reported astrophysical flux in the 100 TeV - PeV range at the level of $10^{-8}\, \mathrm{GeV}\, \mathrm{cm}^{-2}\, \mathrm{s}^{-1}\, \mathrm{sr}^{-1}$ per flavor and reject a purely atmospheric explanation for the combined 3-year data at $5.7 \sigma$. The data are consistent with expectations for equal fluxes of all three neutrino flavors and with isotropic arrival directions, suggesting either numerous or spatially extended sources. The three-year dataset, with a livetime of 988 days, contains a total of 37 neutrino candidate events with deposited energies ranging from 30 to 2000 TeV. The 2000 TeV event is the highest-energy neutrino interaction ever observed.