  Correlated Basis Function (CBF) perturbation theory and the formalism of
cluster expansions have been recently employed to obtain an effective
interaction from a nuclear Hamiltonian strongly constrained by phenomenology.
We report the results of a study of the superfluid gap in pure neutron matter,
associated with the formation of Cooper pairs in the $^1S_0$ channel. The
calculations have been carried out using an improved version of the CBF
effective interaction, in which three-nucleon forces are taken into account
using a microscopic model. Our results show that a non-vanishing superfluid gap
develops at densities in the range $2 \times 10^{-4} \lesssim \rho/\rho_0
\lesssim 0.1 $, where $\rho_0 = 2.8 \times 10^{14}$ g cm$^{-3}$ is the
equilibrium density of isospin-symmetric nuclear matter, corresponding mainly
to the neutron star inner crust.
