A dual-band chiral metasurface structure with giant optical activity and negative refractivity is proposed, which is composed of a periodic array of a middle dielectric layer and two conjugated gammadion layers. The reasons for the occurrences of giant optical activity and negative refractive indexes are clarified by analyzing the surface current density distributions. The effects of the radius of the connected circular metallic patch in the unit structure of the chiral metasurface and the thickness of the dielectric layer on optical activity and negative refractivity are studied. The numerical simulation results show that there are four resonance frequencies within the frequency range from 0.1 THz to 2 THz, and near these resonance frequencies, the average refractive index is negative and the maximal magnitude of the real part is -3.7. The proposed structure indicates its exceptional giant optical activity as well as its dual-band negative refractivity for a left circularly polarized wave and a right circularly polarized wave near these resonance frequencies. The maximum polarization rotation angle reaches 122° and the magnitude of the real part of the refractive index of a right circularly polarized wave reaches -12.74.