A surface force balance with extremely high resolution in measuring shear forces has been used to study the properties of films of the simple organic solvents cyclohexane, octamethylcyclotetrasiloxane, and toluene, confined in a gap between smooth solid surfaces. We were able to probe in detail the transition between liquidlike and solidlike behavior of the films as the gap thickness decreased. Our results reveal that in such confined layers the liquids are fluid down to a film thickness of few molecular layers (typically seven, depending on the particular liquid examined). On further decreasing the gap thickness by a single molecular layer, the films undergo an abrupt transition to become solidlike in the sense that they are able to sustain a finite sheer stress for macroscopic times. At the transition, the effective rigidity of the films, quantified in terms of an effective creep viscosity, increases by at least seven orders of magnitude. This sharp transition is reversible and occurs as a function of the confinement alone : it does not require external applied pressure. Following the transition the confined films behave under shear in a manner resembling ductile solids.