We investigate the mixing of a stratified fluid of finite volume by a turbulent buoyant plume. We develop a model to describe the mixing and apply this to both the cases of a two-layer stratification and a continuous stratification. With a two-layer stratification, the plume intrudes at the interface where it supplies an intermediate layer of fluid. This new layer gradually deepens, primarily mixing the original near-source layer of fluid through entrainment. Eventually, this intermediate layer becomes sufficiently buoyant that the plume penetrates into the more distal layer, leaving a partially mixed region between the original layers of fluid. Analysis of new experiments shows that the growth of the intermediate layer depends primarily on the ratio λ of (i) the filling box time, during which the plume entrains a volume of fluid equal to that in the near-source layer, and (ii) the time for the buoyancy of the near-source layer to increase to that of the more distal layer. For small values of λ, the near-source layer becomes approximately well mixed, and the penetration time of the plume scales with the buoyancy evolution time of the near-source layer. In the limit λ ~ O(1), however, the plume penetrates through into the distal layer long before the near-source layer becomes well mixed; instead, at the time of penetration, the plume leaves an intermediate partially mixed zone between the two original layers. We develop a new phenomenological model to account for the mixing in this intermediate layer based on the effective turbulent diffusion associated with the kinetic energy in the plume and compare this with the model for penetrative entrainment proposed by Kumagai (J. Fluid Mech., vol. 147, 1984, p. 105). In comparison with the experimental data, the models provide a reasonably accurate prediction of the plume penetration time, while the diffusive mixing model provides a somewhat more accurate description of the evolution of the density profile for a range 0 < λ < 1. The diffusive mixing model also leads to predictions which are consistent with some new experimental data for the case in which a plume mixes a continuously stratified layer. In particular, the model is able to predict the initial transient mixing of the region between the source and the height at which the plume intrudes laterally in the ambient fluid, thereby providing an advance on the late-time mixing model of Cardoso and Woods (J. Fluid Mech., vol. 250, 1993, p. 277). We consider the implications of these results on the turbulent penetrative entrainment associated with buoyant plumes.