Generative AI appears poised to transform white collar professions, with more than 90% of Fortune 500 companies using OpenAI's flagship GPT models, which have been characterized as "general purpose technologies" capable of effecting epochal changes in the economy. But how will such technologies impact organizations whose job is to verify and report factual information, and to ensure the health of the information ecosystem? To investigate this question, we conducted 30 interviews with N=38 participants working at 29 fact-checking organizations across six continents, asking about how they use generative AI and the opportunities and challenges they see in the technology. We found that uses of generative AI envisioned by fact-checkers differ based on organizational infrastructure, with applications for quality assurance in Editing, for trend analysis in Investigation, and for information literacy in Advocacy. We used the TOE framework to describe participant concerns ranging from the Technological (lack of transparency), to the Organizational (resource constraints), to the Environmental (uncertain and evolving policy). Building on the insights of our participants, we describe value tensions between fact-checking and generative AI, and propose a novel Verification dimension to the design space of generative models for information verification work. Finally, we outline an agenda for fairness, accountability, and transparency research to support the responsible use of generative AI in fact-checking. Throughout, we highlight the importance of human infrastructure and labor in producing verified information in collaboration with AI. We expect that this work will inform not only the scientific literature on fact-checking, but also contribute to understanding of organizational adaptation to a powerful but unreliable new technology.