  We measure the cross-correlation between weak lensing of galaxy images and of
the cosmic microwave background (CMB). The effects of gravitational lensing on
different sources will be correlated if the lensing is caused by the same mass
fluctuations. We use galaxy shape measurements from 139 deg$^{2}$ of the Dark
Energy Survey (DES) Science Verification data and overlapping CMB lensing from
the South Pole Telescope (SPT) and Planck. The DES source galaxies have a
median redshift of $z_{\rm med} {\sim} 0.7$, while the CMB lensing kernel is
broad and peaks at $z{\sim}2$. The resulting cross-correlation is maximally
sensitive to mass fluctuations at $z{\sim}0.44$. Assuming the Planck 2015
best-fit cosmology, the amplitude of the DES$\times$SPT cross-power is found to
be $A = 0.88 \pm 0.30$ and that from DES$\times$Planck to be $A = 0.86 \pm
0.39$, where $A=1$ corresponds to the theoretical prediction. These are
consistent with the expected signal and correspond to significances of $2.9
\sigma$ and $2.2 \sigma$ respectively. We demonstrate that our results are
robust to a number of important systematic effects including the shear
measurement method, estimator choice, photometric redshift uncertainty and CMB
lensing systematics. Significant intrinsic alignment of galaxy shapes would
increase the cross-correlation signal inferred from the data; we calculate a
value of $A = 1.08 \pm 0.36$ for DES$\times$SPT when we correct the
observations with a simple IA model. With three measurements of this
cross-correlation now existing in the literature, there is not yet reliable
evidence for any deviation from the expected LCDM level of cross-correlation,
given the size of the statistical uncertainties and the significant impact of
systematic errors, particularly IAs. We provide forecasts for the expected
signal-to-noise of the combination of the five-year DES survey and SPT-3G.
