We have detected the intrinsic size of Sagittarius A*, the Galactic Center radio source associated with a supermassive black hole, showing that the short-wavelength radio emission arises from very near the event horizon of the black hole. Radio observations with the Very Long Baseline Array show that the source has a size of \(24 \pm 2\) Schwarzschild radii at 7 mm wavelength. In one of eight 7-mm epochs we also detect an increase in the intrinsic size of \(60^{+25}_{-17}\)%. These observations place a lower limit to the mass density of Sgr A* of \(1.4\times10^{4}\) solar masses per cubic astronomical unit.