We present galaxy-galaxy lensing results from 139 deg(2) of Dark Energy Survey (DES) Science Verification (SV) data. Our lens sample consists of red galaxies, known as redMaGiC, which are specifically selected to have a low photometric redshift error and outlier rate. The lensing measurement has a total signal-to-noise ratio of 29 over scales 0.09 < R < 15 Mpc h(-1), including all lenses over a wide redshift range 0.2 < z < 0.8. Dividing the lenses into three redshift bins for this constant moving number density sample, we find no evidence for evolution in the halo mass with redshift. We obtain consistent results for the lensing measurement with two independent shear pipelines, NGMIX and IM3SHAPE. We perform a number of null tests on the shear and photometric redshift catalogues and quantify resulting systematic uncertainties. Covariances from jackknife subsamples of the data are validated with a suite of 50 mock surveys. The result and systematic checks in this work provide a critical input for future cosmological and galaxy evolution studies with the DES data and redMaGiC galaxy samples. We fit a halo occupation distribution (HOD) model, and demonstrate that our data constrain the mean halo mass of the lens galaxies, despite strong degeneracies between individual HOD parameters.