  We present a cosmological analysis using the second and third moments of the
weak lensing mass (convergence) maps from the first three years of data (Y3)
data of the Dark Energy Survey (DES). The survey spans an effective area of
4139 square degrees and uses the images of over 100 million galaxies to
reconstruct the convergence field. The second moment of the convergence as a
function of smoothing scale contains information similar to standard shear
2-point statistics. The third moment, or the skewness, contains additional
non-Gaussian information. The data is analysed in the context of the
$\Lambda$CDM model, varying 5 cosmological parameters and 19 nuisance
parameters modelling astrophysical and measurement systematics. Our modelling
of the observables is completely analytical, and has been tested with
simulations in our previous methodology study. We obtain a 1.7\% measurement of
the amplitude of fluctuations parameter $S_8\equiv \sigma_8
(\Omega_m/0.3)^{0.5} = 0.784\pm 0.013$. The measurements are shown to be
internally consistent across redshift bins, angular scales, and between second
and third moments. In particular, the measured third moment is consistent with
the expectation of gravitational clustering under the $\Lambda$CDM model. The
addition of the third moment improves the constraints on $S_8$ and $\Omega_{\rm
m}$ by $\sim$15\% and $\sim$25\% compared to an analysis that only uses second
moments. We compare our results with {\it Planck} constraints from the Cosmic
Microwave Background (CMB), finding a $2.2$ \textendash $2.8\sigma$ tension in
the full parameter space, depending on the combination of moments considered.
The third moment independently is in $2.8\sigma$ tension with {\it Planck}, and
thus provides a cross-check on analyses of 2-point correlations.
