We use 26×106 galaxies from the Dark Energy Survey (DES) Year 1 shape catalogs over 1321 deg2 of the sky to produce the most significant measurement of cosmic shear in a galaxy survey to date. We constrain cosmological parameters in both the flat ΛCDM and the wCDM models, while also varying the neutrino mass density. These results are shown to be robust using two independent shape catalogs, two independent photo-z calibration methods, and two independent analysis pipelines in a blind analysis. We find a 3.5% fractional uncertainty on σ8(Ωm/0.3)0.5=0.782-0.027+0.027 at 68% C.L., which is a factor of 2.5 improvement over the fractional constraining power of our DES Science Verification results. In wCDM, we find a 4.8% fractional uncertainty on σ8(Ωm/0.3)0.5=0.777-0.038+0.036 and a dark energy equation-of-state w=-0.95-0.39+0.33. We find results that are consistent with previous cosmic shear constraints in σ8 - Ωm, and we see no evidence for disagreement of our weak lensing data with data from the cosmic microwave background. Finally, we find no evidence preferring a wCDM model allowing w≠-1. We expect further significant improvements with subsequent years of DES data, which will more than triple the sky coverage of our shape catalogs and double the effective integrated exposure time per galaxy.