Ice-penetrating radar profiles reveal distinctive internal structure within the flat ice terrains that bound the lower reaches of the now stagnant Kamb Ice Stream, West Antarctica; slightly warped but continuous upper layers overlie extensive deep line diffractors located at a uniform depth several hundreds of meters above the bed. On the basis of their orientation and morphology we interpret the line diffractors to be the tips of basal crevasses. At the boundary between these terrains and the inter--ice stream ridges, internal layers are strongly downwarped, and the deepest layers are truncated at the bed. Results from simple kinematic ice flow modeling indicate that the observed layer pattern requires a cumulative basal melt of 120--350 m over a time period that is less than 300 years. Fields of basal crevasses are typically observed in ice shelves; however, isolated basal crevasses can also be found in grounded ice when the subglacial water pressure is close to overburden. On the basis of the presence of these crevasses we argue that these terrains were once floating or near floating in the past. We combine model results with estimates of recent ice thickness change to suggest that floatation was transitory and the region may have become fully grounded roughly 200 years prior to stagnation of Kamb Ice Stream.