From two-dimensional ground magnetic and ionospheric electric field observations it should in principle be possible to estimate the two-dimensional distribution of the ionospheric conductance. In the past this was achieved using auroral zone data by modeling the ionospheric conductances on a trial-and-error basis until the observed magnetic variations were reproduced. No information about the existence or the uniqueness of the solution was available this way. We have developed a method to directly deduce the Hall conductance distribution &Sgr;H from ground magnetic and ionospheric electric field observations based upon some assumption for the ratio of the Hall to the Pedersen conductivity. In general, the solutions are shown not to be unique because for a specific solution the value of &Sgr;H on certain parts of the boundary of the two-dimensional domain has to be specified. However, in many situations, especially in the presence of strong and isolated field-aligned currents, these boundary values become less influential, and the solution can be shown to be practically unique over a large area of the domain. In some cases, a rather restrictive relation between the electric field and the equivalent height-integrated current density is shown to hold that could be used to cross-check the quality of the observations. As a specific example, we apply our formalism to the observation of a Harang discontinuity obtained in northern Scandinavia simultaneously by the STARE coherent radar system and the IMS Scandinavian Magnetometer Array. ¿ American Geophysical Union 1992