  We measure the stacked lensing signal in the direction of galaxy clusters in
the Dark Energy Survey Year 3 (DES Y3) redMaPPer sample, using cosmic microwave
background (CMB) temperature data from SPT-3G, the third-generation CMB camera
on the South Pole Telescope (SPT). We estimate the lensing signal using
temperature maps constructed from the initial 2 years of data from the SPT-3G
'Main' survey, covering 1500 deg$^2$ of the Southern sky. We then use this
signal as a proxy for the mean cluster mass of the DES sample. In this work, we
employ three versions of the redMaPPer catalogue: a Flux-Limited sample
containing 8865 clusters, a Volume-Limited sample with 5391 clusters, and a
Volume&Redshift-Limited sample with 4450 clusters. For the three samples, we
find the mean cluster masses to be ${M}_{200{\rm{m}}}=1.66\pm0.13$
[stat.]$\pm0.03$ [sys.], $1.97\pm0.18$ [stat.]$\pm0.05$ [sys.], and
$2.11\pm0.20$ [stat.]$\pm0.05$ [sys.]$\times{10}^{14}\ {\rm{M}}_{\odot }$,
respectively. This is a factor of $\sim2$ improvement relative to the precision
of measurements with previous generations of SPT surveys and the most
constraining cluster mass measurements using CMB cluster lensing to date.
Overall, we find no significant tensions between our results and masses given
by redMaPPer mass-richness scaling relations of previous works, which were
calibrated using CMB cluster lensing, optical weak lensing, and velocity
dispersion measurements from various combinations of DES, SDSS and Planck data.
We then divide our sample into 3 redshift and 3 richness bins, finding no
significant tensions with optical weak-lensing calibrated masses in these bins.
We forecast a $5.7\%$ constraint on the mean cluster mass of the DES Y3 sample
with the complete SPT-3G surveys when using both temperature and polarization
data and including an additional $\sim1400$ deg$^2$ of observations from the
'Extended' SPT-3G survey.
