We present spectroscopic observations of V404 Cygni obtained during the decline from a major out-burst. We follow the spectral evolution over the period from 1989 July to 1990 May and find a spectrum characteristic of emission from an accretion disk in a close binary system. V404 Cyg displays a smooth continuum, lacking in stellar absorption features, with broad, double-peaked Balmer, He, and Fe emission lines whose velocity widths (FWHM approximately 600-1400 km s-1) increased with time. We interpret the increasing line widths as evidence for a decrease in the outer radius of the disk's emission-line region. Time-resolved spectroscopy sets an upper limit on the semiamplitude of the projected velocity variation of the compact object of K1 less-than-or-similar-to 25 km s-1, for orbital periods less-than-or-equal-to 6 hr. A classical mass determination is not possible, but if the emission lines originate in the outer disk, as their profiles suggest, then for the most likely range of the orbital period (4-6 hr), the binary inclination i is less-than-or-similar-to 20-degrees, and the mass of the compact star M1 greater-than-or-similar-to 1.4 M..