Models are presented of the electrodynamic interaction between corotating Jovian magnetospheric plasma and the ionosphere of Io. The interaction is treated as a purely transverse Alfv¿n wave which is controlled by the electrical conductivity distribution of Io's ionosphere. By quantifying the connection between the ionospheric conductivity and the magnetic field and velocity perturbations observed during the Voyager 1 flyby, the observations serve as a probe of the ionosphere, under the assumption that the ionosphere constitutes the principal cross-field conductance near Io. If the electron density is assumed to be that measured earlier by Pioneer 10, the ionospheric resistivity is controlled by the electron-neutral collision rate. Parameterized approximations for the neutral density distribution are varied to produce a fit between the calculated magnetic field and observations made by Voyager 1. The best-fitting ionospheric models assume a neutral density distribution containing a significant global component (of the order of 109 to 1010 cm-3). However, the cold-trap model (according to which Io has a very low-density atmosphere) can be accommodated with the assumption of a significant SO2 coverage of the trailing hemisphere resulting from outgassing of the major plumes there. After taking account of field line curvature and Alfv¿n speed variation over the distance between Io and Voyager 1, the Io plasma torus charged-mass density required for a proper match to the data is larger than earlier estimates based on the width of the Alfv¿n wing signature.