The adsorption of C60 molecules on the Cu(111)1X1 surface has been investigated by field ion-scanning tunneling microscopy (FI-STM). At the initial stage adsorption, C60 molecules are mobile on the terrace at room temperature and segregate to the steps to form linear chains. With increasing coverage, two-dimensional islands form with a close-packed configuration. Upon monolayer adsorption, a highly ordered Cu(111)-(4X4)-C60 overlayer forms. Bias voltage-dependent scanning tunneling microscope images of the individual C60 molecules show unique intramolecular structures. The adsorption geometry of C60 is determined by analyzing the scanning tunneling microscope images and the intramolecular structures are interpreted as the local density of the states of C60 interacting with the substrate.