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Neural networks are powerful tools. Applying them in computer systems—operating systems, databases, and networked systems—attracts much attention. However, neural networks are complicated black boxes that may produce unexpected results. To train networks with well-defined behaviors, we introduce ouroboros, a system that constructs verified neural networks. Verified neural networks are those that satisfy user-defined safety properties, known as specifications. Ouroboros builds verified networks by a training-verification loop that combines deep learning training and neural network verification. The system employs multiple techniques to fill the gap between today’s verification and the properties required for systems. Ouroboros also accelerates the training-verification loop by spec-aware learning. Our experiments show that ouroboros can train verified networks for five applications that we study and has a 2.8× speedup on average compared with the vanilla training-verification loop.