The educational landscape is transforming due to new skill demands, fragmented attention spans, evolving communication patterns, and increased information access. Game-based learning (GBL) has emerged as a potential solution, fostering subject competency and broader skills. However, integrating games into education faces challenges for educators and schools. This study examines Generative AI's (GenAI) role in facilitating board game use in education, based on a case study in an Italian high school implementing a GenAI-enhanced board game for foreign language teaching. Two state-of-the-art GenAI technologies assisted in the whole GBL design process. Findings indicate GenAI can streamline board game integration through flexible rule adaptation, customized artifact production, and rapid content creation, reducing educator workload, as well as extend their capabilities with novel, creative solutions. Benefits are more pronounced when GenAI use is based on an overarching educational strategy. Study's findings have implications for educators and researchers seeking to effectively integrate GBL and GenAI in educational settings.