Neutron diffraction, magnetic susceptibility, specific heat, and dielectric constant measurements of single-crystal Co3TeO6 have been measured to study the interplay between the ferroelectricity and magnetic order. Long-range incommensurate magnetic order develops below T-M1 = 26 K, which is followed by three additional zero-field phase transitions at T-M2 = 19.5 K, T-M3 = 18 K, and T-M4 = 16 K where the incommensurate order changes and commensurate order develops. In magnetic fields up to 14 T we find that the magnetic intensities and incommensurate wave vector are dramatically altered as ferroelectricity develops, with a fifth abrupt transition around 10 T. The overall behavior characterizes Co3TeO6 as a type-II multiferroic.