We use weak gravitational lensing measurements from Year 3 Dark Energy Survey data to calibrate the masses of 443 galaxy clusters selected via the Sunyaev-Zel'dovich effect from Atacama Cosmology Telescope Data Release 5 maps of the cosmic microwave background. We incorporate redshift and SZ measurements for individual clusters into a hierarchical model for the stacked lensing signals and perform Bayesian analyses to constrain the hydrostatic mass bias of the clusters. Our treatment of systematic uncertainties includes a prescription for measuring and accounting for the weak lensing boost factor, consideration of a miscentering effect, as well as marginalization over uncertainties in the source galaxy photometric redshift distributions and shear calibration. The resultant constraints on the normalization of the mass-observable relation have a precision of approximately 7%, with the mean WL halo mass of M 500c = 5.4 × 1014 M ⊙. We measure the bias between the true cluster mass and the mass estimated from the SZ signal based on an X-ray--calibrated scaling relation assuming hydrostatic equilibrium, to be 1 - b = 0.74+0.06 -0.05 over the full sample. When splitting the clusters into high (z = 0.43--0.70) and low (z = 0.15--0.43) redshift bins, we measure 1 - b = 0.58+0.06 -0.05 and 0.81+0.08 -0.06, respectively. When introducing additional freedom in redshift and mass to the hydrostatic bias model, we find that 1 - b decreases with redshift (with the power law of -1.8+0.5 -0.6, 99.95% confidence), consistent with findings from other recent studies, while we do not find any significant trend in mass. We also demonstrate that our result is robust against various systematics such as a scale cut, priors on baryonic and miscentering parameters, and degree of scatter in mass-observable relation. The weak-lensing mass calibration presented in this study will be a useful tool for using the ACT clusters as probes of astrophysics, and as a step towards using their abundance as a cosmological probe.