The interaction of methanethiol (CH3SH) molecules with the Cu(111) surface was investigated using a first-principles method based on density functional theory, and a slab model. A series of possible adsorption configurations constructed using S atoms on different sites with different tilt angles were studied. It was found for the first time that the non-dissociative molecular adsorption of CH3SH on the Cu(111) surface with the S atom sitting on the top site belongs to the weak chemisorption, and the adsorption energy is 0.39 eV. After the dissociation of the S - H bond, the S atom is located at the bridge site, with a small shift toward the hollow site. The dissociative adsorption structure is thermodynamically more stable than the intact one, and the adsorption energy is 0.75-0.77 eV. Two reaction pathways have been studied for the transition from non-dissociative adsorption to dissociative adsorption, and the activation energy barrier along the minimum energy path is 0.57 eV. The results of the calculations indicated that the released H atom prefers to form a bond with the copper surface, rather than desorbing in the H-2 molecular form. Comparing the local density of states of S atoms in the single CH3SH, CH3SH/Cu(111), and CH3S/Cu(111) structures, we found that the bonding between the S atoms and the substrate is much stronger in the dissociated adsorption states.