The behavior of molecularly thin prototypical confined films [Lennard-Jones (12,6) fluid constrained between two plane-parallel walls consisting of like atoms fixed in the fee (100) configuration] is studied by Monte Carlo in a new (grand isostress) ensemble whose parameters are the thermodynamic state variables [temperature T, chemical potential mu, and normal stress (load) applied to the walls T-zz] controlled in the surface forces apparatus used to study lubrication experimentally on a molecular scale. Additional parameters of the ensemble not generally controlled in this experiment are the film-wall interfacial area A and the crystallographic alignment (registry, or shear strain OL) Of the walls. A multiplicity of phases is found to comport with a given choice of the parameters. The thermodynamically stable one minimizes the grand isostress potential (free energy). By means of thermodynamic integration the stable phase of the film is determined as a function of ct at fixed T, mu, T-zz, and A. Solid films comprising integral numbers of layers of atoms parallel with the walls are stable when the walls are appropriately aligned. When such films are sheared (alpha is varied), they undergo drainage (imbibition) transitions at critical strains, whereupon the him loses (gains) whole layers. Depending on the conditions, the solid film may melt just prior to the transition.