  The $^1S_0$ pairing in neutron matter has been investigated in presence of
realistic two-- and three--nucleon interactions. We have adopted the Argonne
$v_{8^\prime}$ NN and the Urbana IX 3N potentials. Quantum Monte Carlo theory,
specifically the Auxiliary Field Diffusion Monte Carlo method, and Correlated
Basis Function theory are employed in order to get quantitative and reliable
estimates of the gap. They both fully take into account the medium
modifications due to the interaction induced correlations. The two methods are
in good agreement up to the maximum gap density and both point to a slight
reduction with respect to the standard BCS value. In fact, the maximum gap is
about $2.5 \text{MeV}$ at $k_F \sim 0.8 \text{fm}^{-1}$ in BCS and 2.3--$2.4
\text{MeV}$ at $k_F \sim 0.6 \text{fm}^{-1}$ in correlated matter. At higher
densities the Quantum Monte Carlo gap becomes close to BCS. In general, the
computed medium polarization effects are much smaller than those previously
estimated within \emph{all theories}. Truncations of Argonne $v_{8^\prime}$ to
simpler forms give the same gaps in BCS, provided the truncated potentials have
been refitted to the same NN data set. Differences among the models appear in
the correlated theories, most of the reduction being attributable to the tensor
force. The three--nucleon interaction provides an additional increase of the
gap of about 0.35 MeV.
