  Over forty years of research suggests that the common envelope phase, in
which an evolved star engulfs its companion upon expansion, is the critical
evolutionary stage forming short-period, compact-object binary systems, such as
coalescing double compact objects, X-ray binaries, and cataclysmic variables.
In this work, we adapt the one-dimensional hydrodynamic stellar evolution code,
MESA, to model the inspiral of a 1.4M$_{\odot}$ neutron star (NS) inside the
envelope of a 12$M_{\odot}$ red supergiant star. We self-consistently calculate
the drag force experienced by the NS as well as the back-reaction onto the
expanding envelope as the NS spirals in. Nearly all of the hydrogen envelope
escapes, expanding to large radii ($\sim$10$^2$ AU) where it forms an optically
thick envelope with temperatures low enough that dust formation occurs. We
simulate the NS orbit until only 0.8M$_{\odot}$ of the hydrogen envelope
remains around the giant star's core. Our results suggest that the inspiral
will continue until another $\approx$0.3M$_{\odot}$ are removed, at which point
the remaining envelope will retract. Upon separation, a phase of dynamically
stable mass transfer onto the NS accretor is likely to ensue, which may be
observable as an ultraluminous X-ray source. The resulting binary, comprised of
a detached 2.6M$_{\odot}$ helium-star and a NS with a separation of
3.3-5.7R$_{\odot}$, is expected to evolve into a merging double neutron-star,
analogous to those recently detected by LIGO/Virgo. For our chosen combination
of binary parameters, our estimated final separation (including the phase of
stable mass transfer) suggests a very high $\alpha_{\rm CE}$-equivalent
efficiency of $\approx$5.
