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The AC Optimal Power Flow (AC-OPF) is a key building block in many power system applications. It determines generator setpoints at minimal cost that meet the power demands while satisfying the underlying physical and operational constraints. It is non-convex and NP-hard, and computationally challenging for large-scale power systems. Motivated by the increased stochasticity in generation schedules and increasing penetration of renewable sources, this paper explores a deep learning approach to deliver highly efficient and accurate approximations to the AC-OPF. In particular, the paper proposes an integration of deep neural networks and Lagrangian duality to capture the physical and operational constraints. The resulting model, called OPF-DNN, is evaluated on real case studies from the French transmission system, with up to 3,400 buses and 4,500 lines. Computational results show that OPF-DNN produces highly accurate AC-OPF approximations whose costs are within 0.01% of optimality. OPF-DNN generates, in milliseconds, solutions that capture the problem constraints with high fidelity.