In the wake of its accretion, the Moon was likely partially or fully molten, forming a lunar magma ocean (LMO). A fully molten Moon may crystallize neutrally buoyant olivine, whose accumulation may form a barrier leading to a separation into two independently evolving melt reservoirs. This study investigates the crystallization of such a putative basal lunar magma ocean, ranging from the core mantle boundary (with radius r = 380 km) to the level of neutral olivine buoyancy (r = 600 km). Consecutive crystallization experiments determine liquidus temperatures and crystallize 10–30 wt% minerals, before conceptually segregating these according to their buoyancy, and then stepping to a new bulk composition that corresponds to the residual melt after removal of the cumulate minerals.The first olivine to crystallize from a Taylor Whole Moon composition (twm, XMg = Mg/(Mg + Fe2+) = 0.83) has XMg = 0.94 and is neutrally buoyant at 3.8 GPa according to the melt density model of Lange and Carmichael (1990). Crystallization begins at the core mantle boundary, but early olivine floats and re-dissolves until the magma ocean cools to the liquidus temperature at the depth of neutral buoyancy (1850 °C). At this point a > 500 km thick olivine-only layer could form, mainly fed from the upper magma shell. The crystallization sequence in the basal magma ocean is olivine-only at 1850–1675 °C, 0–26 pcs (wt% percent solidified) → olivine + opx (to 1600 °C, 42 pcs) → opx + cpx + garnet (to 1580 °C, 60 pcs) → cpx + garnet (to 1520 °C, 73 pcs) → cpx + garnet + olivine leaving 11 wt% residual melt at 1450 °C.Cooling this last melt to 1300 °C leads to 80% garnet + cpx + olivine + Ti-spinel, the residual liquid corresponding to 2.2% of the basal magma ocean. Olivine, opx and cpx remain buoyant over the entire crystallization interval and various pyroxenite layers are added to the olivine layer. Garnet and the final cotectic cpx + garnet + olivine + FeTi-oxide assemblage instead form a 70 km thick basal layer on the core-mantle boundary. Such a layer provides a gravitationally stable high-density lowermost lunar mantle, which would concur with a recent re-analysis of the lunar seismic data. The lowest temperature experiment at 1300 °C, not much above the 1250 °C proposed for the core-mantle boundary from inversion of geophysical data, had about 20% of a highly evolved, dense Fe-rich melt. It is feasible that this melt has remained in its liquid state to present day, providing an explanation for the proposed low vp and vs layer and high dissipation just above the core-mantle boundary.