First-principles calculations reveal a large spin-phonon coupling in cubic SrMnO3, with ferromagnetic ordering producing a polar instability. Through combination of this coupling with the strain-polarization coupling characteristic of perovskites, the bulk antiferromagnetic-paraelectric ground state is driven to a previously unreported multiferroic ferroelectric-ferromagnetic state by increasing epitaxial strain. This state has a computed P-s > 54 mu C/cm(2) and magnetic T-c > 92 K. Large mixed magnetic-electric-elastic responses are predicted in the vicinity of the phase boundaries