We used a numerical ice-sheet model to reconstruct the North American Laurentide Ice Sheet during the last glacial maximum. Our model simulates ice-sheet conditions that can be specified experimentally as either a rigid substrate (hard bed) or a wet, deformable till (soft bed); basal sliding is excluded. We use geologic records of former basal ice-sheet processes to prescribe the distribution of hard and soft beds. Our reconstruction of the Laurentide Ice Sheet is significantly lower in ice-surface height and contains less ice volume than the CLIMAP (maximum) reconstruction. In contrast, our reconstruction agrees well with the ICE-4G reconstruction, both in height and volume. Because the ICE-4G reconstruction is based on the inversion of relative sea-level data, whereas our reconstruction is based on glacial geology and ice mechanics, this agreement suggests that soft beds provide a glaciological mechanism to explain the shape and volume of the Laurentide Ice Sheet that is most consistent with observations of relative sea-level change and other geodynamic considerations.