The tip of a low-temperature scanning tunneling microscope is approached towards a C-60 molecule adsorbed at a pentagon-hexagon bond on Cu(100) to form a tip-molecule contact. The conductance rapidly increases to approximate to 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 C-60. The conductance in the transition region is affected by structural fluctuations which modulate the tip-molecule distance.