We propose a novel technique for guidance of buoyancy-controlled vehicles in uncertain under-ice ocean flows. In-situ melt rate measurements collected at the grounding zone of Antarctic ice shelves, where the ice shelf meets the underlying bedrock, are essential to constrain models of future sea level rise. Buoyancy-controlled vehicles, which control their vertical position in the water column through internal actuation but have no means of horizontal propulsion, offer an affordable and reliable platform for such in-situ data collection. However, reaching the grounding zone requires vehicles to traverse tens of kilometers under the ice shelf, with approximate position knowledge and no means of communication, in highly variable and uncertain ocean currents. To address this challenge, we propose a partially observable MDP approach that exploits model-based knowledge of the under-ice currents and, critically, of their uncertainty, to synthesize effective guidance policies. The approach uses approximate dynamic programming to model uncertainty in the currents, and QMDP to address localization uncertainty. Numerical experiments show that the policy can deliver up to 88.8% of underwater vehicles to the grounding zone -- a 33% improvement compared to state-of-the-art guidance techniques, and a 262% improvement over uncontrolled drifters. Collectively, these results show that model-based under-ice guidance is a highly promising technique for exploration of under-ice cavities, and has the potential to enable cost-effective and scalable access to these challenging and rarely observed environments.