Control of magnetic domain walls (DWs) and their propagation is among the most promising development directions for future information-storage devices. The well-established tools for such manipulation are the spin-torque transfer from electrical currents and strain. The focus of this paper is an alternative concept based on the nonvolatile ferroelectric field effect on DWs in a ferromagnet with carrier-mediated exchange coupling. The integrated ferromagnet/ferroelectric structure yields two superimposed ferroic patterns strongly coupled by an electric field. Using this coupling, we demonstrate an easy-to-form, stable, nondestructive, and electrically rewritable switch on magnetic domain wall propagation.