  Cross-correlation between weak lensing of the Cosmic Microwave Background
(CMB) and weak lensing of galaxies offers a way to place robust constraints on
cosmological and astrophysical parameters with reduced sensitivity to certain
systematic effects affecting individual surveys. We measure the angular
cross-power spectrum between the Atacama Cosmology Telescope (ACT) DR4 CMB
lensing and the galaxy weak lensing measured by the Dark Energy Survey (DES) Y3
data. Our baseline analysis uses the CMB convergence map derived from ACT-DR4
and $\textit{Planck}$ data, where most of the contamination due to the thermal
Sunyaev Zel'dovich effect is removed, thus avoiding important systematics in
the cross-correlation. In our modelling, we consider the nuisance parameters of
the photometric uncertainty, multiplicative shear bias and intrinsic alignment
of galaxies. The resulting cross-power spectrum has a signal-to-noise ratio $=
7.1$ and passes a set of null tests. We use it to infer the amplitude of the
fluctuations in the matter distribution ($S_8 \equiv \sigma_8 (\Omega_{\rm
m}/0.3)^{0.5} = 0.782\pm 0.059$) with informative but well-motivated priors on
the nuisance parameters. We also investigate the validity of these priors by
significantly relaxing them and checking the consistency of the resulting
posteriors, finding them consistent, albeit only with relatively weak
constraints. This cross-correlation measurement will improve significantly with
the new ACT-DR6 lensing map and form a key component of the joint 6x2pt
analysis between DES and ACT.
