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A promising approach to improve climate‐model simulations is to replace traditional subgrid parameterizations based on simplified physical models by machine learning algorithms that are data‐driven. However, neural networks (NNs) often lead to instabilities and climate drift when coupled to an atmospheric model. Here, we learn an NN parameterization from a high‐resolution atmospheric simulation in an idealized domain by accurately calculating subgrid terms through coarse graining. The NN parameterization has a structure that ensures physical constraints are respected, such as by predicting subgrid fluxes instead of tendencies. The NN parameterization leads to stable simulations that replicate the climate of the high‐resolution simulation with similar accuracy to a successful random‐forest parameterization while needing far less memory. We find that the simulations are stable for different horizontal resolutions and a variety of NN architectures, and that an NN with substantially reduced numerical precision could decrease computational costs without affecting the quality of simulations.