  We obtain exact scattering solutions of the Dirac equation in 1+1 dimensions
for a double square barrier vector potential. The potential floor between the
two barriers is higher than 2mc^2 whereas the top of the barriers is at least
2mc^2 above the floor. The relativistic version of the conventional double
barrier transmission resonance is obtained for energies within + or - mc^2 from
the height of the barriers. However, we also find two more (sub-barrier)
transmission resonance regions below the conventional one. Both are located
within the two Klein energy zones and characterized by resonances that are
broader than the conventional ones.
