  Uncertainty in the mass-observable scaling relations is currently the
limiting factor for galaxy cluster based cosmology. Weak gravitational lensing
can provide a direct mass calibration and reduce the mass uncertainty. We
present 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 them
with previously reported space-based observations of 13 galaxy clusters at
redshifts $0.576 \leq z \leq 1.132$ to constrain the cluster mass scaling
relations 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 previously
used framework for the analysis of scaling relations and cosmological
constraints obtained from SPT-selected clusters to make use of weak lensing
information. We introduce a new approach to estimate the effective average
redshift distribution of background galaxies and quantify a number of
systematic errors affecting the weak lensing modelling. These errors include a
calibration of the bias incurred by fitting a Navarro-Frenk-White profile to
the reduced shear using $N$-body simulations. We blind the analysis to avoid
confirmation bias. We are able to limit the systematic uncertainties to 5.6% in
cluster mass (68% confidence). Our constraints on the mass--X-ray observable
scaling relations parameters are consistent with those obtained by earlier
studies, and our constraints for the mass--SZE scaling relation are consistent
with the simulation-based prior used in the most recent SPT-SZ cosmology
analysis. We can now replace the external mass calibration priors used in
previous SPT-SZ cosmology studies with a direct, internal calibration obtained
on the same clusters.
