Monte Carlo simulations of the adsorption of potassium on a Cu(111) surface at 80 K and coverages going from theta=0.13 ＇ to theta=0.26 monolayer are reported. The particles interact through an adsorbate-substrate potential previously derived and through a classical adsorbate-adsorbate Morse-type potential. Interesting results concerning the nature of the adsorbed phase have been obtained : (a) We have found that the potassium adlayer is incommensurate under 1 ML coverage (theta=0.25), but it becomes commensurate at 1 ML, forming a p(2X2) adlayer with respect to the Cu substrate. (b) A procedure to determine the distribution of potassium on different adsorption sites in terms of theta has been developed with the result that top sites are the preferred sites for adsorption, although at low and medium coverages other sites are also occupied. At 1 ML only the top sites are occupied, in good agreement with the available experimental data.