Neural Network Potentials (NNPs) have quickly emerged as powerful computational methods for modeling large chemical systems with the accuracy of quantum mechanical methods but at a much smaller computational cost. To make the training and evaluation of the underlying neural networks practical, these methods commonly cutoff interatomic interactions at a modest range (e.g., 5~\AA), so longer-range interactions like London dispersion are neglected. This limits the accuracy of these models for intermolecular interactions. In this work, we develop a new NNP designed for modeling chemical systems were dispersion is an essential component. This new NNP is extended to treat dispersion interactions rigorously by calculating atomic dispersion coefficients through a second NN, which is trained to reproduce the coefficients from the quantum-mechanically derived exchange-hole dipole moment (XDM) model. Calculation of the dispersion component of intermolecular interactions through this scheme provides results in very good agreement with the QM data, with a mean absolute error (MAE) of 0.6 kcal/mol and a coefficient of determination (R2) of 0.98. The dispersion components of these intermolecular interactions are predicted in excellent agreement with the QM data, with a mean absolute error (MAE) of 0.02 kcal/mol and an R2 of 1.00. This combined dispersion-corrected NNP, called ANIPBE0-MLXDM, predicts intermolecular interaction energies for complexes from the DE370K test set with an MAE of 0.5 kcal/mol and an R2 of 0.94 relative to high-level ab initio results (CCSD(T)/CBS), but with a computational cost that is billions of times smaller. The ANIPBE0-MLXDM method is effective for simulating large-scale dispersion-driven systems, like gas adsorption in porous materials, molecular liquids, and nanostructures, on a single computer workstation.