A Lennard-Jones fluid layer adsorbed on a smooth solid surface was studied at coverages $ heta$ ~ 0.8 to 1.8 on an isotherm by performing intensive grand canonical Monte Carlo simulations. The results clearly show a picture of two-step melting process which used to be observed in recent thermodynamic measurements of argon monolayer melting on graphite. The observed melting process consist of an abrupt density change followed by a gradual transition. Snapshots of monolayer configurations indicate that the creation and dissociation of a dislocation pair are involved in the melting mechanism. Taking the effect of system size into account, it is suggested that, while the abrupt density change may be not related to the theory of Kosterlitz, Thouless, Halperin, Nelson, and Young (KTHNY), the second gradual transition is probably a KTHNY-type melting transition.