  This is the fourth in a series of papers studying the astrophysics and
cosmology of massive, dynamically relaxed galaxy clusters. Here, we use
measurements of weak gravitational lensing from the Weighing the Giants project
to calibrate Chandra X-ray measurements of total mass that rely on the
assumption of hydrostatic equilibrium. This comparison of X-ray and lensing
masses provides a measurement of the combined bias of X-ray hydrostatic masses
due to both astrophysical and instrumental sources. Assuming a fixed cosmology,
and within a characteristic radius (r_2500) determined from the X-ray data, we
measure a lensing to X-ray mass ratio of 0.96 +/- 9% (stat) +/- 9% (sys). We
find no significant trends of this ratio with mass, redshift or the
morphological indicators used to select the sample. In accordance with
predictions from hydro simulations for the most massive, relaxed clusters, our
results disfavor strong, tens-of-percent departures from hydrostatic
equilibrium at these radii. In addition, we find a mean concentration of the
sample measured from lensing data of c_200 = $3.0_{-1.8}^{+4.4}$. Anticipated
short-term improvements in lensing systematics, and a modest expansion of the
relaxed lensing sample, can easily increase the measurement precision by
30--50%, leading to similar improvements in cosmological constraints that
employ X-ray hydrostatic mass estimates, such as on Omega_m from the cluster
gas mass fraction.
