The response of grounded ice sheets to a changing climate critically influences possible future changes in sea level. Recent satellite surveys over southern Greenland show little overall elevation change at higher elevations, but large spatial variability. Using satellite studies alone, it is not possible to determine the geophysical processes responsible for the observed elevation changes and to decide if recent rates of change exceed the natural variability. Here we derive changes in ice-sheet elevation in southern Greenland, for the years 1978-88, using a physically based model of firn densification and records of annual snow accumulation reconstructed from 12 ice cores at high elevation. Our patterns of accumulation-driven elevation change agree closely with contemporaneous satellite measurements of ice-sheet elevation change, and we therefore attribute the changes observed in 1978-88 to variability in snow accumulation. Similar analyses of longer ice-core records show that in this decade the Greenland ice sheet exhibited typical variability at high elevations, well within the long-term natural variability. Our results indicate that a better understanding of ice-sheet mass changes will require long-term measurements of both surface elevation and snow accumulation.