Translating satellite measurements of ice sheet volume change into sea level contribution requires knowledge of the profile of density as a function of depth within the ice sheet, and how this profile changes over time. This paper describes an interferometric method of inverting ground-penetrating radar returns for the profile of firn density as a function of depth. The method is an interferometric implementation of the common-midpoint approach, performed using a stepped-frequency, phase-sensitive ground-penetrating radar. By recording the phase difference of returns with a range of antenna separations, the different path lengths through the firn allow recovery of a smoothed representation of the density profile. This density model is characterised by three parameters: surface density and two decay lengths for porosity, each operating over a different density range. Our results suggest that the stepped-frequency radar used here can accurately recover differences in two-way travel time and produce useful estimates of the density profile. In a test of the method performed at Summit station in Greenland, the recovered density-depth profile agreed with indepenent density measurements from an ice core and a neutron probe to within 6% root-mean-square error.