The electronic structure of isolated bis(phthalocyaninato) terbium(III) molecules, a novel single-molecular-magnet (SMM), supported on the Cu(111) surface has been characterized by density functional theory and scanning tunneling spectroscopy. These studies reveal that the interaction with the metal surface preserves both the molecular structure and the large spin magnetic moment of the metal center. The 4f electron states are not perturbed by the adsorption while a strong molecular/metal interaction can induce the suppression of the minor spin contribution delocalized over the molecular ligands. The calculations show that the inherent spin magnetic moment of the molecule is only weakly affected by the interaction with the surface and suggest that the SMM character might be preserved.