  We study a gas of fermions undergoing a wide resonance s-wave BCS-BEC
crossover, in the BEC regime at zero temperature. We calculate the chemical
potential and the speed of sound of this Bose-condensed gas, as well as the
condensate depletion, in the low density approximation. We discuss how higher
order terms in the low density expansion can be constructed. We demonstrate
that the standard BCS-BEC gap equation is invalid in the BEC regime and is
inconsistent with the results obtained here. We indicate how our theory can in
principle be extended to nonzero temperature. The low density approximation we
employ breaks down in the intermediate BCS-BEC crossover region. Hence our
theory is unable to predict how the chemical potential and the speed of sound
evolve once the interactions are tuned towards the BCS regime. As a part of our
theory, we derive the well known result for the bosonic scattering length
diagrammatically and check that there are no bound states of two bosons.
