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Many companies provide neural network prediction services to users for a wide range of applications. However, current prediction systems compromise one party's privacy: either the user has to send sensitive inputs to the service provider for classification, or the service provider must store its proprietary neural networks on the user's device. The former harms the personal privacy of the user, while the latter reveals the service provider's proprietary model. We design, implement, and evaluate Delphi, a secure prediction system that allows two parties to execute neural network inference without revealing either party's data. Delphi approaches the problem by simultaneously co-designing cryptography and machine learning. We first design a hybrid cryptographic protocol that improves upon the communication and computation costs over prior work. Second, we develop a planner that automatically generates neural network architecture configurations that navigate the performance-accuracy trade-offs of our hybrid protocol. Together, these techniques allow us to achieve a 22x improvement in online prediction latency compared to the state-of-the-art prior work.