Rotating superfluid 3He-B is found to possess a new contribution to the NMR frequency shift, which changes sign on reversal of either the angular velocity of rotation or the magnetic field. For p=29.3 bars this gyromagnetic effect shows a discontinuity in magnitude at the first-order phase-transition temperature T/Tc=0.6, at which a change in the vortex-core structure takes place. These observations support the conclusion that the vortex core possesses a spontaneous intrinsic magnetization.