The integration of Artificial Intelligence (AI) into safety-critical systems introduces a new reliability paradigm: silent failures, where AI produces confident but incorrect outputs that can be dangerous. This paper introduces the Formal Assurance and Monitoring Environment (FAME), a novel framework that confronts this challenge. FAME synergizes the mathematical rigor of offline formal synthesis with the vigilance of online runtime monitoring to create a verifiable safety net around opaque AI components. We demonstrate its efficacy in an autonomous vehicle perception system, where FAME successfully detected 93.5% of critical safety violations that were otherwise silent. By contextualizing our framework within the ISO 26262 and ISO/PAS 8800 standards, we provide reliability engineers with a practical, certifiable pathway for deploying trustworthy AI. FAME represents a crucial shift from accepting probabilistic performance to enforcing provable safety in next-generation systems.