  We present the first constraints on cosmology from the Dark Energy Survey
(DES), using weak lensing measurements from the preliminary Science
Verification (SV) data. We use 139 square degrees of SV data, which is less
than 3\% of the full DES survey area. Using cosmic shear 2-point measurements
over three redshift bins we find $\sigma_8 (\Omega_{\rm m}/0.3)^{0.5} = 0.81
\pm 0.06$ (68\% confidence), after marginalising over 7 systematics parameters
and 3 other cosmological parameters. We examine the robustness of our results
to the choice of data vector and systematics assumed, and find them to be
stable. About $20$\% of our error bar comes from marginalising over shear and
photometric redshift calibration uncertainties. The current state-of-the-art
cosmic shear measurements from CFHTLenS are mildly discrepant with the
cosmological constraints from Planck CMB data; our results are consistent with
both datasets. Our uncertainties are $\sim$30\% larger than those from CFHTLenS
when we carry out a comparable analysis of the two datasets, which we attribute
largely to the lower number density of our shear catalogue. We investigate
constraints on dark energy and find that, with this small fraction of the full
survey, the DES SV constraints make negligible impact on the Planck
constraints. The moderate disagreement between the CFHTLenS and Planck values
of $\sigma_8 (\Omega_{\rm m}/0.3)^{0.5}$ is present regardless of the value of
$w$.
