Shear peak statistics has gained a lot of attention recently as a practical alternative to the two point statistics for constraining cosmological parameters. We perform a shear peak statistics analysis of the Dark Energy Survey (DES) Science Verification (SV) data, using weak gravitational lensing measurements from a 139 deg2 field. We measure the abundance of peaks identified in aperture mass maps, as a function of their signal-to-noise ratio, in the signal-to-noise range 0 ă S{N ă 4. To predict the peak counts as a function of cosmological parameters we use a suite of N-body simulations spanning 158 models with varying Ωm and σ8, fixing w “ ́1, Ωb “ 0.04, h “ 0.7 and ns “ 1, to which we have applied the DES SV mask and redshift distribution. In our fiducial analysis we measure σ8 pΩm {0.3q0.6 “ 0.77 ̆ 0.07, after marginalising over the shear multiplicative bias and the error on the mean redshift of the galaxy sample. We introduce models of intrinsic alignments, blending, and source contamination by cluster members. These models indicate that peaks with S{N ą 4 would require significant corrections, which is why we do not include them in our analysis. We compare our results to the cosmological constraints from the two point analysis on the SV field and find them to be in good agreement in both the central value and its uncertainty. We discuss prospects for future peak statistics analysis with upcoming DES data.