Infrared reflectivity spectra of cubic SrMnO3 ceramics reveal 18% stiffening of the lowest-frequency phonon below the antiferromagnetic phase transition occurring at T-N = 233 K. Such a large temperature change of the polar phonon frequency is extraordinary and we attribute it to an exceptionally strong spin-phonon coupling in this material. This is consistent with our prediction from first-principles calculations. Moreover, polar phonons become Raman active below T-N, although their activation is forbidden by symmetry in the Pm (3) over barm space group. This gives evidence that the cubic Pm (3) over barm symmetry is locally broken below T-N due to a strong magnetoelectric coupling. Multiphonon and multimagnon scattering is also observed in Raman spectra. Microwave and THz permittivity is strongly influenced by hopping electronic conductivity, which is caused by small nonstoichiometry of the sample. Thermoelectric measurements show room-temperature concentration of free carriers n(e) = 3.6 x 10(20) cm(-3) and the sample composition Sr2+Mn0.984+Mn0.023+O2.992-. The conductivity exhibits very unusual temperature behavior: THz conductivity increases on cooling, while the static conductivity markedly decreases on cooling. We attribute this to different conductivity of the ceramic grains and grain boundariesclose