We show here that mixed-valence (MV) magnetic molecules with a significant electron delocalization are extremely sensitive to an external electric field. In particular, we focus on the symmetric many-electron MV binuclear complexes that are on the borderline between Robin and Day classes II and III. In these molecules, the double-exchange, which has been shown to lead to the ferromagnetic ground spin state, competes with the electric field, which tends to localize the spin, thus creating an electric dipole and stabilizing the spin states with lower multiplicities. This provides an efficient and easy way to control the ground spin state of the molecule through the double-exchange mechanism. Thus, we predict that the application of an external electric field will lead to a strong stepwise decrease of the magnetic susceptibility and to a simultaneous increase of the electric polarization. The reverse effect, consisting of a sharp decrease of the electric polarization under the action of an external magnetic field, is also predicted. The results demonstrate that MV dimers of this class can be regarded as single-molecule analogs of multiferroics with promising potential to create a functional magnetoelectric unit in one molecule.