The introduction of Hydrogen (H) into Zirconium (Zr) influences many mechanical properties, especially due to low H solubility and easy formation of Zirconium hydride phases. Understanding the various effects of H requires studies with atomistic resolution but at scales that incorporate defects such as cracks, interfaces, and dislocations. Such studies thus demand accurate interatomic potentials. Here, a neural network potential (NNP) for the Zr-H system is developed within the Behler-Parrinello framework. The Zr-H NNP retains the accuracy of a recent NNP for hcp Zr and exhibits excellent agreement with first-principles density functional theory (DFT) for (i) H interstitials and their diffusion in hcp Zr, (ii) formation energies, elastic constants, and surface energies of relevant Zr hydrides, and (iii) energetics of a common Zr/Zr-H interface. The Zr-H NNP shows physical behavior for many different crack orientations in the most-stable ε-hydride, and structures and reasonable relative energetics for the 〈a〉 screw dislocation in pure Zr. This Zr-H NNP should thus be very powerful for future study of many phenomena driving H degradation in Zr that require atomistic detail at scales far above those accessible by first-principles.