We report that an external electric field applied normal to bilayers of transition-metal dichalcogenides TX2, M = Mo, W, X = S, Se, creates significant spin-orbit splittings and reduces the electronic band gap linearly with the field strength. Contrary to the TX2 monolayers, spin-orbit splittings and valley polarization are absent in bilayers due to the presence of inversion symmetry. This symmetry can be broken by an electric field, and the spin-orbit splittings in the valence band quickly reach similar values as in the monolayers (145 meV for MoS2... 418 meV for WSe2) at saturation fields less than 500 mV A-1. The band gap closure results in a semiconductor-metal transition at field strength between 1.25 (WX2) and 1.50 (MoX2) V A-1. Thus, by using a gate voltage, the spin polarization can be switched on and off in TX2 bilayers, thus activating them for spintronic and valleytronic applications.