The thickness of ice shelves, a basic parameter for mass balance estimates, is typically inferred using hydrostatic equilibrium for which knowledge of the depth-averaged density is essential. The densification from snow to ice depends on a number of local factors (e.g. temperature and surface mass balance) causing spatial and temporal variations in density–depth profiles. However, direct measurements of firn density are sparse, requiring substantial logistical effort. Here, we infer density from radio-wave propagation speed using ground-based wide-angle radar datasets (10 MHz) collected at five sites on Roi Baudouin Ice Shelf (RBIS), Dronning Maud Land, Antarctica. Using a novel algorithm including traveltime inversion and raytracing with a prescribed shape of the depth–density relationship, we show that the depth to internal reflectors, the local ice thickness and depth-averaged densities can reliably be reconstructed. For the particular case of an ice-shelf channel, where ice thickness and surface slope change substantially over a few kilometers, the radar data suggests that firn inside the channel is about 5 % denser than outside the channel. Although this density difference is at the detection limit of the radar, it is consistent with a similar density anomaly reconstructed from optical televiewing, which reveals 10 % denser firn inside compared to outside the channel. The denser firn in the ice-shelf channel should be accounted for when using the hydrostatic ice thickness for determining basal melt rates. The radar method presented here is robust and can easily be adapted to different radar frequencies and data-acquisition geometries.