The thickness and equivalent global sea level contribution of an improved model of the central and northern Laurentide Ice Sheet is constrained by 24 relative sea level histories and 18 present-day GPS-measured vertical land motion rates. The final model, termed Laur16, is derived from the ICE-5G model by holding the timing history constant and iteratively adjusting the thickness history, in four regions of northern Canada. In the final model, the last glacial maximum (LGM) thickness of the Laurentide Ice Sheet west of Hudson Bay was ~3.4-3.6 km. Conversely, east of Hudson Bay, peak ice thicknesses reached ~4 km. The ice model thicknesses inferred for these two regions represent, respectively, a ~30 per cent decrease and an average ~20-25 per cent increase to the load thickness relative to the ICE-5G reconstruction, which is generally consistent with other recent studies that have focussed on Laurentide Ice Sheet history. The final model also features peak ice thicknesses of 1.2-1.3 km in the Baffin Island region, a modest reduction relative to ICE-5G and unchanged thicknesses for a region in the central Canadian Arctic Archipelago west of Baffin Island. Vertical land motion predictions of the final model fit observed crustal uplift rates well, after an adjustment is made for the elastic crustal response to present-day ice mass changes of regional ice cover. The new Laur16 model provides more than a factor of two improvement of the fit to the RSL data (?2 measure of misfit) and a factor of nine improvement to the fit of the GPS data (mean squared error measure of fit), compared to the ICE-5G starting model. Laur16 also fits the regional RSL data better by a factor of two and gives a slightly better fit to GPS uplift rates than the recent ICE-6G model. The volume history of the Laur16 reconstruction corresponds to an up to 8 m reduction in global sea level equivalent compared to ICE-5G at LGM.