The transition from rest to sliding contact of atomically smooth solids separated by molecularly thin liquid films was studied. The films could be deformed nearly reversibly to a large fraction of the film thickness. The modulus of elasticity and yield stress were low, considerably less than for a molecular crystal or glass in the bulk. The transition to dissipative sliding was typically (but not always) discontinuous. The dissipative stress was then nearly velocity-independent. The similar response of monolayers strongly attached to the solid surfaces, presenting a well-defined interface for sliding, suggests that the physical mechanism of sliding may involve wall slip.