  We report results of the equation of state of neutron matter in the
low--density regime, where the Fermi wave vector ranges from $0.4 fm^{-1} \leq
k_F \leq 1.0 fm^{-1}$. Neutron matter in this regime is superfluid because of
the strong and attractive interaction in the $^1S_0$ channel. The properties of
this superfluid matter are calculated starting from a realistic Hamiltonian
that contains modern two-- and three--body interactions. The ground state
energy and the $^1S_0$ superfluid energy gap are calculated using the Auxiliary
Field Diffusion Monte Carlo method. We study the structure of the ground state
by looking at pair distribution functions as well as the Cooper-pair wave
function used in the calculations.
