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 stars 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.
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