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Model predictive control (MPC) can provide significant energy cost savings in building operations in the form of energy-efficient control with better occupant comfort, lower peak demand charges, and risk-free participation in demand response. However, the engineering effort required to obtain physics-based models of buildings is considered to be the biggest bottleneck in making MPC scalable to real buildings. In this paper, we propose a data-driven control algorithm based on neural networks to reduce this cost of model identification. Our approach does not require building domain expertise or retrofitting of existing heating and cooling systems. We validate our learning and control algorithms on a two-story building with ten independently controlled zones, located in Italy. We learn dynamical models of energy consumption and zone temperatures with high accuracy and demonstrate energy savings and better occupant comfort compared to the default system controller.