The conductance of single alkanedithiols covalently bound to gold electrodes has been studied by statistical analysis of repeatedly created molecular junctions. For each molecule, the conductance histogram reveals two sets of well-defined peaks, corresponding to two different conductance values. We have found that (1) both conductance values decrease exponentially with the molecular length with an identical decay constant, beta approximate to 0.84 angstrom(-1), but with a factor of 5 difference in the prefactor of the exponential function. (2) The current-voltage curves of the two sets can be fit with the Simmons tunneling model. (3) Both conductance values are independent of temperature (between -5 and 60 degrees C) and the solvent. (4) Despite the difference in the conductance, the forces required to break the molecular junctions are the same, 1.5 nN. These observations lead us to believe that the conduction mechanism in alkanedithiols is due to electron tunneling or superexchange via the bonds along the molecules, and the two sets of conductance peaks are due to two different microscopic configurations of the molecule-electrode contacts.