We use 26 million galaxies from the Dark Energy Survey ( DES ) Year 1 shape catalogs over 1321 deg ^ { 2 } 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 \Lambda CDM and w CDM 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 \sigma _ { 8 } ( \Omega _ { m } / 0.3 ) ^ { 0.5 } = 0.782 ^ { +0.027 } _ { -0.027 } at 68 % CL , which is a factor of 2.5 improvement over the fractional constraining power of our DES Science Verification results . In w CDM , we find a 4.8 % fractional uncertainty on \sigma _ { 8 } ( \Omega _ { m } / 0.3 ) ^ { 0.5 } = 0.777 ^ { +0.036 } _ { -0.038 } and a dark energy equation-of-state w = -0.95 ^ { +0.33 } _ { -0.39 } . We find results that are consistent with previous cosmic shear constraints in \sigma _ { 8 } – \Omega _ { m } , and see no evidence for disagreement of our weak lensing data with data from the CMB . Finally , we find no evidence preferring a w CDM model allowing w \neq - 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 .