Young adults today face unprecedented mental health challenges, yet many hesitate to seek support due to barriers such as accessibility, stigma, and time constraints. Bite-sized well-being interventions offer a promising solution to preventing mental distress before it escalates to clinical levels, but have not yet been delivered through personalized, interactive, and scalable technology. We conducted the first multi-institutional, longitudinal, preregistered randomized controlled trial of a generative AI-powered mobile app ("Flourish") designed to address this gap. Over six weeks in Fall 2024, 486 undergraduate students from three U.S. institutions were randomized to receive app access or waitlist control. Participants in the treatment condition reported significantly greater positive affect, resilience, and social well-being (i.e., increased belonging, closeness to community, and reduced loneliness) and were buffered against declines in mindfulness and flourishing. These findings suggest that, with purposeful and ethical design, generative AI can deliver proactive, population-level well-being interventions that produce measurable benefits.