It is generally understood that small-scale mixing, such as is caused by breaking internal waves, drives upwelling of the densest ocean waters that sink to the ocean bottom at high latitudes. However, the observational evidence that the strong turbulent fluxes generated by small-scale mixing in the stratified ocean interior are more vigorous close to the ocean bottom boundary than above implies that small-scale mixing converts light waters into denser ones, thus driving a net sinking of abyssal waters. Using a combination of theoretical ideas and numerical models, it is argued that abyssal waters upwell along weakly stratified boundary layers, where small-scale mixing of density decreases to zero to satisfy the no density flux condition at the ocean bottom. The abyssal ocean meridional overturning circulation is the small residual of a large net sinking of waters, driven by small-scale mixing in the stratified interior above the bottom boundary layers, and a slightly larger net upwelling, driven by the decay of small-scale mixing in the boundary layers. The crucial importance of upwelling along boundary layers in closing the abyssal overturning circulation is the main finding of this work.