The first evidence of successful p-type doping of InN is presented. It is shown that InN:Mg films consist of a p-type bulk region with a thin n-type inversion layer at the surface that prevents electrical contact to the bulk. Capacitance-voltage measurements indicate a net concentration of ionized acceptors below the -type surface. Irradiation with 2 MeV He+ ions is used to convert the bulk of InN:Mg from p to n-type, at which point photoluminescence is recovered. The conversion is well explained by a model assuming two parallel conducting layers (the surface and the bulk) in the films.