An experiment has been completed which demonstrated quantum mechanical tunneling of electrons between two gold electrodes separated in vacuum. The tunneling current between the gold electrodes has been measured, for fixed voltages of 0.1 and 0.01 volts, as the electrode spacing was varied from a distance of approximately 2.0 nm down to a point where the electrodes touched. Current-voltage characteristics for fixed electrode spacing in the direct tunneling region have also been measured. Numerical calculations of the tunneling current based on the free-electron model of the electrodes and the barrier, an image-potential reduced barrier, and a WKB approximation for the tunneling probability have been performed and compared with Simmons' theory and with the experimental results. Within experimental error the results indicate that an image potential reduced barrier with the modifications suggested by Lang and Kohn gives a close approximation to the true barrier for metal-vacuum-metal tunneling. For the first time, the work function of the electrodes in a tunneling experiment has been deduced from experimental parameters independent of the tunneling device.