A new model of the last deglaciation event of the Late Quaternary ice age is here described and denoted as ICE-6G_C (VM5a). It differs from previously published models in this sequence in that it has been explicitly refined by applying all of the available Global Positioning System (GPS) measurements of vertical motion of the crust that may be brought to bear to constrain the thickness of local ice cover as well as the timing of its removal. Additional space geodetic constraints have also been applied to specify the reference frame within which the GPS data are described. The focus of the paper is upon the three main regions of Last Glacial Maximum ice cover, namely, North America, Northwestern Europe/Eurasia, and Antarctica, although Greenland and the British Isles will also be included, if peripherally, in the discussion. In each of the three major regions, the model predictions of the time rate of change of the gravitational field are also compared to that being measured by the Gravity Recovery and Climate Experiment satellites as an independent means of verifying the improvement of the model achieved by applying the GPS constraints. Several aspects of the global characteristics of this new model are also discussed, including the nature of relative sea level history predictions at far-field locations, in particular the Caribbean island of Barbados, from which especially high-quality records of postglacial sea level change are available but which records were not employed in the development of the model. Although ICE-6G_C (VM5a) is a significant improvement insofar as the most recently available GPS observations are concerned, comparison of model predictions with such far-field relative sea level histories enables us to identify a series of additional improvements that should follow from a further stage of model iteration.