Anomalous coarse-grained sediment layers beneath the North Atlantic likely originated from sediment freeze-on to the base of ice sheets during the last glacial period. These layers represent periods of extreme ice discharge, called Heinrich events, and are variously attributed to ice stream flow instability, ice shelf collapse, or enhanced terminus melting due to ocean warming. In this paper, we study the processes controlling how sediment freezes on to the base of ice streams and predict the volume of sediment conveyed by icebergs during a Heinrich event. The local thickness of frozen sediment is sensitive to the heat flux at the ice-bed interface and the water pressure, both of which also contribute to the controls on basal friction; as the basal water pressure increases, both the frozen sediment thickness and the basal friction decrease. The sediment discharged during a Heinrich event must have frozen on to the ice during the inter-Heinrich period. As the Heinrich event proceeds, the frozen sediment melts off the base of the ice stream, indicating that the thickness of sediments deposits in the North Atlantic may not reliably constrain Heinrich event duration. Choosing reasonable parameters corresponding to the Hudson Strait Ice Stream, our model of sediment freeze-on and discharge is consistent with observational estimates of Heinrich event sediment discharge volume.