  Number counts of galaxy clusters across redshift are a powerful cosmological
probe, if a precise and accurate reconstruction of the underlying mass
distribution is performed -- a challenge called mass calibration. With the
advent of wide and deep photometric surveys, weak gravitational lensing by
clusters has become the method of choice to perform this measurement. We
measure and validate the weak gravitational lensing (WL) signature in the shape
of galaxies observed in the first 3 years of the DES Y3 caused by galaxy
clusters selected in the first all-sky survey performed by SRG/eROSITA. These
data are then used to determine the scaling between X-ray photon count rate of
the clusters and their halo mass and redshift. We empirically determine the
degree of cluster member contamination in our background source sample. The
individual cluster shear profiles are then analysed with a Bayesian population
model that self-consistently accounts for the lens sample selection and
contamination, and includes marginalization over a host of instrumental and
astrophysical systematics. To quantify the accuracy of the mass extraction of
that model, we perform mass measurements on mock cluster catalogs with
realistic synthetic shear profiles. This allows us to establish that
hydro-dynamical modelling uncertainties at low lens redshifts ($z<0.6$) are the
dominant systematic limitation. At high lens redshift the uncertainties of the
sources' photometric redshift calibration dominate. With regard to the X-ray
count rate to halo mass relation, we constrain all its parameters. This work
sets the stage for a joint analysis with the number counts of eRASS1 clusters
to constrain a host of cosmological parameters. We demonstrate that WL mass
calibration of galaxy clusters can be performed successfully with source
galaxies whose calibration was performed primarily for cosmic shear
experiments.
