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Uncertainty in the mass-observable scaling relations is currently thelimiting factor for galaxy cluster based cosmology. Weak gravitational lensingcan provide a direct mass calibration and reduce the mass uncertainty. Wepresent new ground-based weak lensing observations of 19 South Pole Telescope(SPT) selected clusters at redshifts $0.29 \leq z \leq 0.61$ and combine themwith previously reported space-based observations of 13 galaxy clusters atredshifts $0.576 \leq z \leq 1.132$ to constrain the cluster mass scalingrelations with the Sunyaev-Zel'dovich effect (SZE), the cluster gas mass \mgas,and \yx, the product of \mgas\ and X-ray temperature. We extend a previouslyused framework for the analysis of scaling relations and cosmologicalconstraints obtained from SPT-selected clusters to make use of weak lensinginformation. We introduce a new approach to estimate the effective averageredshift distribution of background galaxies and quantify a number ofsystematic errors affecting the weak lensing modelling. These errors include acalibration of the bias incurred by fitting a Navarro-Frenk-White profile tothe reduced shear using $N$-body simulations. We blind the analysis to avoidconfirmation bias. We are able to limit the systematic uncertainties to 5.6% incluster mass (68% confidence). Our constraints on the mass--X-ray observablescaling relations parameters are consistent with those obtained by earlierstudies, and our constraints for the mass--SZE scaling relation are consistentwith the simulation-based prior used in the most recent SPT-SZ cosmologyanalysis. We can now replace the external mass calibration priors used inprevious SPT-SZ cosmology studies with a direct, internal calibration obtainedon the same clusters.

Sunyaev-Zel'dovich Effect and X-ray Scaling Relations from Weak-Lensing Mass Calibration of 32 SPT Selected Galaxy Clusters