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Spectroscopic techniques have been used to model the V644 Mon (HD 51840) system, a single-line binary consisting of an early-type main sequence (B1 V) star and a mass-losing late-type supergiant (K0 I-II). High resolution CCD spectra of V644 Mon were obtained at the Ritter and McMath-Pierce Observatories between November 1993 and April 1994. An orbital solution yields a period of 120.6$\pm$0.1 days and semi-amplitude for the K star of 60.4$\pm$1.4 km/s. A \mbox{$v \sin i$} \ for the K star of 20$\pm$5 km/s is measured, and its bolometric magnitude and temperature suggest a radius near 80\mbox{$R_\odot$}. From these data an inclination near 40 degrees is estimated for the system. Photometric evidence for ellipsoidal variations suggests that the K star is tidally deformed and transferring matter through the inner Lagrangian point onto and around the B star. A consistent model yields masses of 4\mbox{$M_\odot$} \ and 13\mbox{$M_\odot$} \ for the K and B stars, respectively, the former having lost as much as 10\mbox{$M_\odot$}. This material is most likely responsible for the shell/wind iron peak absorption features observed by IUE against the UV continuum of the B star. Optical shell/wind lines are also observed at H$\alpha$, He I $\lambda$5875, Na I D, and Ca II K.
