We present a velocity dispersion-based mass calibration of the South Pole Telescope Sunyaev-Zeldovich effect survey (SPT-SZ) galaxy cluster sample. Using a homogeneously selected sample of 100 cluster candidates from 720 deg2 of the survey along with 63 velocity dispersion ($sigma_v$) and 16 X-ray Yx measurements of sample clusters, we simultaneously calibrate the mass-observable relation and constrain cosmological parameters. The calibrations using $sigma_v$ and Yx are consistent at the $0.6sigma$ level, with the $sigma_v$ calibration preferring ~16% higher masses. We use the full cluster dataset to measure $sigma_8(Omega_ m/0.27)^{0.3}=0.809pm0.036$. The SPT cluster abundance is lower than preferred by either the WMAP9 or Planck+WMAP9 polarization (WP) data, but assuming the sum of the neutrino masses is $sum m_ u=0.06$ eV, we find the datasets to be consistent at the 1.0$sigma$ level for WMAP9 and 1.5$sigma$ for Planck+WP. Allowing for larger $sum m_ u$ further reconciles the results. When we combine the cluster and Planck+WP datasets with BAO and SNIa, the preferred cluster masses are $1.9sigma$ higher than the Yx calibration and $0.8sigma$ higher than the $sigma_v$ calibration. Given the scale of these shifts (~44% and ~23% in mass, respectively), we execute a goodness of fit test; it reveals no tension, indicating that the best-fit model provides an adequate description of the data. Using the multi-probe dataset, we measure $Omega_ m=0.299pm0.009$ and $sigma_8=0.829pm0.011$. Within a $ u$CDM model we find $sum m_ u = 0.148pm0.081$ eV. We present a consistency test of the cosmic growth rate. Allowing both the growth index $gamma$ and the dark energy equation of state parameter $w$ to vary, we find $gamma=0.73pm0.28$ and $w=-1.007pm0.065$, demonstrating that the expansion and the growth histories are consistent with a LCDM model ($gamma=0.55; ,w=-1$).
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