We present the first double-lined spectroscopic orbital elements for the central binary in the massive triple δ Orionis A. The solutions are based on fits of cross-correlation functions of IUE high-dispersion UV spectra and He I λ6678 profiles. The orbital elements for the primary agree well with previous results, and in particular, we confirm the apsidal advance with a period of 224.5±4.5 yr. We also present tomographic reconstructions of the primary and secondary stars' spectra that confirm the O9.5 II classification of the primary and indicate a B0.5 III type for the secondary. The relative line strengths between the reconstructed spectra suggest magnitude differences of Δm=-2.5log(Fs/Fp)=2.6±0.2 in the UV and Δm=2.5±0.3 at 6678 Å. The widths of the UV cross-correlation functions are used to estimate the projected rotational velocities, Vsini=157±6 and 138±16 km.s–1 for the primary and secondary, respectively (which implies that both the primary and the secondary rotate faster than the orbital motion). We used the spectroscopic results to make a constrained fit of the Hipparcos light curve of this eclipsing binary, and the model fits limit the inclination to the range i=67°-77°. The lower limit corresponds to a near Roche-filling configuration that has an absolute magnitude that is consistent with the photometrically determined distance to Ori OB1b, the Orion Belt cluster in which δ Ori resides. The i=67° solution results in masses of Mp=11.2 and Ms=5.6 M☉, both of which are substantially below the expected masses for stars of their luminosity. The binary may have experienced a mass ratio reversal caused by case A Roche lobe overflow or the system may have suffered extensive mass loss through a binary interaction (perhaps during a common envelope phase) in which most of the primary's mass was lost from the system rather than transferred to the secondary. We also made three-component reconstructions to search for the presumed stationary spectrum of the close visual companion δ Ori Ab (Hei 42 Ab). There is no indication of the spectral lines of this tertiary in the UV spectrum, but a broad and shallow feature is apparent in the reconstruction of He I λ6678 indicative of an early B-type star. The tertiary may be a rapid rotator (Vsini~300 km.s–1) or a spectroscopic binary.