The electrical contact resistance of a vertical binary contact between stainless steel balls with a low mechanical load was investigated. Using a statistical approach, we measured the voltage at which the dielectric breakdown occurs within a thin surface oxide layer and the distribution of the contact resistance. Electrical load-bearing conduction through a thin insulating layer was found to occur through two possible sequential processes. In both cases, once a conduction path is formed, the melting of bridges as in conventional contact theory is involved. This suggests that conduction through an oxide layer with a low mechanical load depends mainly on breakdown-induced bridges. Furthermore, the distribution of such path’s resistance shows the log-normal distribution with a long tail toward high resistance.