We measure the cross-correlation between redMaGiC galaxies selected from the Dark Energy Survey (DES) year 1 data and gravitational lensing of the cosmic microwave background (CMB) reconstructed from South Pole Telescope (SPT) and Planck data over 1289 deg2. When combining measurements across multiple galaxy redshift bins spanning the redshift range of 0.15<z<0.90, we reject the hypothesis of no correlation at 19.9σ significance. When removing small-scale data points where thermal Sunyaev-Zel’dovich signal and nonlinear galaxy bias could potentially bias our results, the detection significance is reduced to 9.9σ. We perform a joint analysis of galaxy-CMB lensing cross-correlations and galaxy clustering to constrain cosmology, finding Ωm=0.276+0.029−0.030 and S8=σ8√Ωm/0.3=0.800+0.090−0.094. We also perform two alternate analyses aimed at constraining only the growth rate of cosmic structure as a function of redshift, finding consistency with predictions from the concordance ΛCDM model. The measurements presented here are part of a joint cosmological analysis that combines galaxy clustering, galaxy lensing and CMB lensing using data from DES, SPT and Planck.