An analytical solution is found for the first time describing the distribution of the fluxoid density as well as the distribution of the current density in the current-carrying state of thin superconducting films with a high edge barrier. The conditions are determined for the establishment of the Meissner, the critical and the resistive state in the film. A new equilibrium structure representing an edge-pinning-controlled critical state in thin-film superconductors is predicted to emerge in the finite range of total current.