Na I D lines in the spectrum of the young binary KH 15D have been analyzed in detail. We find an excess absorption component that may be attributed to foreground interstellar absorption, and to gas possibly associated with the solids in the circumbinary disk. The derived column density is log N_NaI = 12.5 cm^-2, centered on a radial velocity that is consistent with the systemic velocity. Subtracting the likely contribution of the ISM leaves log N_NaI ~ 12.3 cm^-2. There is no detectable change in the gas column density across the "knife edge" formed by the opaque grain disk, indicating that the gas and solids have very different scale heights, with the solids being highly settled. Our data support a picture of this circumbinary disk as being composed of a very thin particulate grain layer composed of millimeter-sized or larger objects that are settled within whatever remaining gas may be present. This phase of disk evolution has been hypothesized to exist as a prelude to the formation of planetesimals through gravitational fragmentation, and is expected to be short-lived if much gas were still present in such a disk. Our analysis also reveals the presence of excess Na I emission relative to the comparison spectrum at the radial velocity of the currently visible star that plausibly arises within the magnetosphere of this still-accreting young star.