Large rifts that open in Antarctic ice shelves are known to be filled by snow accumulations, ice shelf fragments, and sea ice. This work demonstrates how these rifts may also be filled from below, through their interaction with ocean water, by marine ice. The model presented here quantifies both the rate of marine ice accumulation at the top of the rift, which results from melt-driven convection at its sides, and the impact of this process on waters occupying the confined environment of the rift. The results show that such a system could fill the rift with tens of meters of marine ice. In the process the temperature and salinity of the ambient water in the rift evolve through two distinct phases. The first is characterized by rapid change that takes the properties of the ambient water to near-equilibrium values, followed by a second phase of much slower change. The melting rate at the lower part of the walls of the rift emerges as the principal factor influencing many aspects of model behavior. Testing the model against field observations from an Antarctic rift showed it to be robust and successful in reproducing the main observed features, including the presence of a thick layer of supercooled ocean water inside the rift.