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 GeV cm^-2 s^-1 sr^-1 per flavor and reject a purely atmospheric explanation for the combined three-year data at 5.7σ. 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 data set, with a live time 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.