We report a ferroelectric transition driven by the off-centering of magnetic Mn(4+) ions in antiferromagnetic Mott insulators Sr(1-x)Ba(x)MnO(3) with a perovskite structure. As x increases, the perovskite lattice shows the typical soft-mode dynamics, as revealed by the momentum-resolved inelastic x-ray scattering and far-infrared spectroscopy, and the ferroelectricity shows up for x ≥ 0.45. The observed polarization is comparable to that for a prototypical ferroelectric BaTiO(3). We further demonstrate that the magnetic order suppresses the ferroelectric lattice dilation by ∼70% and increases the soft-phonon energy by ∼50%, indicating the largest magnetoelectric effects yet attained.