Surges are common at all the major ice caps in Iceland. Ice masses of gigatons may shift from the upper part of the outlet glacier towards the terminus in a few months, advancing the glacier front by up to several kilometres. The advancing ice front may be up to 100 m thick, increasing the load on crustal rocks correspondingly. We use the observed change in crustal loading during a surge of the western part of the Vatnajökull ice cap, Iceland, during 1993–1995 and the corresponding elastic crustal deformation, surveyed with interferometric synthetic aperture radar, to investigate the material properties of the solid Earth in this region. Crustal subsidence due to the surge reaches ∼75 mm at the edge of the Síðujökull outlet glacier. This signal is mixed with a broad uplift signal of ∼12 mm yr−1, relative to our reference area, caused by the ongoing retreat of Vatnajökull in response to climate change. We disentangle the two signals by linear inversion. Finite element modelling is used to investigate the elastic Earth response of the surge, as well as to confirm that no significant viscoelastic deformation occurred as a consequence of the surge. The modelling leads to estimates of the Young's modulus and Poisson's ratio of the underlying Earth. Comparison between the observed and modelled deformation fields is made using a Bayesian approach that yields the estimate of a probability distribution for each of the free parameters. Residuals indicate a good agreement between models and observations. One-layer elastic models result in a Young's modulus of 43.2–49.7 GPa (95 per cent confidence) and Poisson's ratio of 0–0.27, after removal of outliers. Our preferred model, with two elastic layers, provides a better fit to the whole surge signal. This model consists of a 1-km-thick upper layer with an average Young's modulus of 12.9–15.3 GPa and Poisson's ratio of 0.17, overlying a layer with an average Young's modulus of 67.3–81.9 GPa and Poisson's ratio of 0.25.