The tip of a low-temperature scanning tunneling microscope is approached towards a C60 molecule adsorbed at a pentagon-hexagon bond on Cu(100) to form a tip-molecule contact. The conductance rapidly increases to approximately 0.25 conductance quanta in the transition region from tunneling to contact. Ab-initio calculations within density functional theory and nonequilibrium Green's function techniques explain the experimental data in terms of the conductance of an essentially undeformed C60. The conductance in the transition region is affected by structural fluctuations which modulate the tip-molecule distance.