We report a novel pressure-induced spin-density-wave transition in the superconductor Mo3Sb7 figured our by measurements of the electrical resistivity and magnetization under hydrostatic pressure. The critical temperature of superconducting Mo3Sb7 is found to increase with increasing pressure, from 2.15 K at 0.2 kbar up to 2.37 K at 22 kbar. Above 4.5 kbar, superconductivity exists in parallel with a pressure-induced spin-density wave state, revealed by a sharp jump in the electrical resistivity and a maximum in the magnetization at the phase transition temperature T_SDW. The application of pressure shifts T_SDW to lower temperatures, from 6.6 K at 4.5 kbar down to 6.15 K at 22 kbar. A strong magnetic field dependence of T_SDW and a maximum seen in the magnetization indicate an antiferromagnetic character of T_SDW. The pressure dependence of T_c and T_SDW suggests a competition of the SDW and the superconducting states in this system. ; Comment: 12 pages, 4 figures. submitted to PRL