A tunnel junction device was made by immersing mercury electrodes in an aqueous nitrate solution. The junction conductance was measured at zero bias as the two mercury surfaces were brought together in the solution. Changes in separation between the mercury surfaces were calculated from changes in the junction conductance using a simple model of elastic electron tunneling, due to Simmons. An absolute distance scale was established using the estimated hard-sphere diameter of water as an internal standard. Discrete changes in junction conductance were observed when the metal surfaces were separated by less than about 1 nm. We interpret this behavior to be due to the presence of quasi-equilibrium junction geometries which are themselves due to time-averaged structuring of liquid water near the metal surfaces. The longitudinal structuring in the water was found to decay normal to the metal surface with a characteristic length on the order of the molecular diameter. The time-averaged structures of the liquid water domains appear to be similar to the structure of hexagonal ice Ih and do not resemble hard-sphere packing. At zero bias, there appears to be no strong preference for one type of ordered water structure over another, suggesting that hydrogen bonding is the dominant factor determining structure in the liquid water near the metal surface and not metal-water bonding in this case. Our experimental data are in substantial agreement with recent molecular dynamics and Monte Carlo simulations and with analytic theory. There are significant differences between our results for ordering of liquid water at metal surfaces and the results reported previously for local ordering of liquid water in the mica/water/mica surface force-balance apparatus.