  This paper models the orbital inspiral of a neutron star (NS) through the
envelope of its giant-branch companion during a common envelope (CE) episode.
These CE episodes are necessary to produce close pairs of NSs that can inspiral
and merge due to gravitational wave losses in less than a Hubble time. Because
cooling by neutrinos can be very efficient, NSs have been predicted to
accumulate significant mass during CE events, perhaps enough to lead them to
collapse to black holes. We revisit this conclusion with the additional
consideration of CE structure, particularly density gradients across the
embedded NS's accretion radius. This work is informed by our recent numerical
simulations that find that the presence of a density gradient strongly limits
accretion by imposing a net angular momentum to the flow around the NS. Our
calculations suggest that NSs should survive CE encounters. They accrete only
modest amounts of envelope material, $\lesssim 0.1M_\odot$, which is broadly
consistent with mass determinations of double NS binaries. With less mass gain,
NSs must spiral deeper to eject their CE, leading to a potential increase in
mergers. The survival of NSs in CE events has implications for the formation
mechanism of observed double NS binaries, as well as for predicted rates of NS
binary gravitational wave inspirals and their electromagnetic counterparts.
