The use of tunneling signals to sequence DNA is presently hampered by the small tunnel conductance of a junction spanning an entire DNA molecule. The design of a readout system that uses a shorter tunneling path requires knowledge of the absolute conductance across base pairs. We have exploited the stochastic switching of hydrogen-bonded DNA base-nucleoside pairs trapped in a tunnel junction to determine the conductance of individual molecular pairs. This conductance is found to be sensitive to the geometry of the junction, but a subset of the data appears to come from unstrained molecular pairs. The conductances determined from these pairs are within a factor of two of the predictions of density functional calculations. The experimental data reproduces the counterintuitive theoretical prediction that guanine-deoxycytidine pairs (3 H-bonds) have a smaller conductance than adenine-thymine pairs (2 H-bonds). A bimodal distribution of switching lifetimes shows that both H-bonds and molecule-metal contacts break.