  We report weak-lensing masses for 51 of the most X-ray luminous galaxy
clusters known. This cluster sample, introduced earlier in this series of
papers, spans redshifts 0.15 < z_cl < 0.7, and is well suited to calibrate mass
proxies for current cluster cosmology experiments. Cluster masses are measured
with a standard `color-cut' lensing method from three-filter photometry of each
field. Additionally, for 27 cluster fields with at least five-filter
photometry, we measure high-accuracy masses using a new method that exploits
all information available in the photometric redshift posterior probability
distributions of individual galaxies. Using simulations based on the COSMOS-30
catalog, we demonstrate control of systematic biases in the mean mass of the
sample with this method, from photometric redshift biases and associated
uncertainties, to better than 3%. In contrast, we show that the use of
single-point estimators in place of the full photometric redshift posterior
distributions can lead to significant redshift-dependent biases on cluster
masses. The performance of our new photometric redshift-based method allows us
to calibrate `color-cut` masses for all 51 clusters in the present sample to a
total systematic uncertainty of ~7% on the mean mass, a level sufficient to
significantly improve current cosmology constraints from galaxy clusters. Our
results bode well for future cosmological studies of clusters, potentially
reducing the need for exhaustive spectroscopic calibration surveys as compared
to other techniques, when deep, multi-filter optical and near-IR imaging
surveys are coupled with robust photometric redshift methods.
