Generative Artificial Intelligence (GenAI) presents transformative opportunities for organizations, yet both midsize organizations and larger enterprises face distinctive adoption challenges. Midsize organizations encounter resource constraints and limited AI expertise, while enterprises struggle with organizational complexity and coordination challenges. Existing technology adoption frameworks, including TAM (Technology Acceptance Model), TOE (Technology Organization Environment), and DOI (Diffusion of Innovations) theory, lack the specificity required for GenAI implementation across these diverse contexts, creating a critical gap in adoption literature. This paper introduces FAIGMOE (Framework for the Adoption and Integration of Generative AI in Midsize Organizations and Enterprises), a conceptual framework addressing the unique needs of both organizational types. FAIGMOE synthesizes technology adoption theory, organizational change management, and innovation diffusion perspectives into four interconnected phases: Strategic Assessment, Planning and Use Case Development, Implementation and Integration, and Operationalization and Optimization. Each phase provides scalable guidance on readiness assessment, strategic alignment, risk governance, technical architecture, and change management adaptable to organizational scale and complexity. The framework incorporates GenAI specific considerations including prompt engineering, model orchestration, and hallucination management that distinguish it from generic technology adoption frameworks. As a perspective contribution, FAIGMOE provides the first comprehensive conceptual framework explicitly addressing GenAI adoption across midsize and enterprise organizations, offering actionable implementation protocols, assessment instruments, and governance templates requiring empirical validation through future research.