  We present measurements of weak gravitational lensing cosmic shear two-point
statistics using Dark Energy Survey Science Verification data. We demonstrate
that our results are robust to the choice of shear measurement pipeline, either
ngmix or im3shape, and robust to the choice of two-point statistic, including
both real and Fourier-space statistics. Our results pass a suite of null tests
including tests for B-mode contamination and direct tests for any dependence of
the two-point functions on a set of 16 observing conditions and galaxy
properties, such as seeing, airmass, galaxy color, galaxy magnitude, etc. We
furthermore use a large suite of simulations to compute the covariance matrix
of the cosmic shear measurements and assign statistical significance to our
null tests. We find that our covariance matrix is consistent with the halo
model prediction, indicating that it has the appropriate level of halo sample
variance. We compare the same jackknife procedure applied to the data and the
simulations in order to search for additional sources of noise not captured by
the simulations. We find no statistically significant extra sources of noise in
the data. The overall detection significance with tomography for our highest
source density catalog is 9.7sigma. Cosmological constraints from the
measurements in this work are presented in a companion paper (DES et al. 2015).
