A magnetohydrodynamic (MHD) fluid description is typically employed to study the magnetized plasma comprising the solar atmosphere. This approach has had many successes in modeling and explaining solar phenomena. Most often, the plasma is assumed to be fully ionized. While this approach is justified in the higher atmosphere, i.e., the solar corona; the temperature in the lower solar atmosphere is such that a large proportion of the fluid may be electrically neutral. This begs the question: to what degree are the results derived from a fully ionized MHD description valid? In this article, we investigate the effect of partial ionization on buoyancy-driven MHD waves (the MHD analog of internal gravity waves) by applying a simplified two-fluid description. We show that previously derived results may be applied, when the fluid is weakly ionized, if the ion-neutral collision frequency is high. We derive dispersion relations for buoyancy-driven MHD waves, which include correction factors and damping rates due to ion-neutral collisions.