Model routing dynamically directs prompts to appropriate AI models to optimize performance and cost. Current static routing methods are limited by an information deficit, unable to learn from execution outcomes. Agent-as-a-Router, a new framework, treats routing as a dynamic, memory-building agent using a Context-Action-Feedback loop. This loop tracks model successes and failures to continuously update the router's behavior. ACRouter, a practical implementation, significantly outperforms static routers and expensive default strategies. It adapts to changes in user behavior and foundation models without requiring extensive model training or complex rules. Static routers fail because they lack execution feedback, cannot adapt to new data, and become obsolete with model updates. Agent-as-a-Router overcomes this by accumulating execution-grounded information during deployment. The C-A-F loop enables the router to learn from past interactions and improve future routing decisions. ACRouter leverages modules for memory, orchestration, and verification, supported by a tool layer for real-world execution feedback. Benchmarks show ACRouter achieves high accuracy and cost savings across diverse tasks, including complex out-of-distribution scenarios. The framework is best suited for verifiable tasks and domains where different models excel in distinct niches.