Sagittarius A*, the ~4 × 10[superscript 6] M ☉ black hole candidate at the Galactic center, can be studied on Schwarzschild radius scales with (sub)millimeter wavelength very long baseline interferometry (VLBI). We report on 1.3 mm wavelength observations of Sgr A* using a VLBI array consisting of the JCMT on Mauna Kea, the Arizona Radio Observatory's Submillimeter Telescope on Mt. Graham in Arizona, and two telescopes of the CARMA array at Cedar Flat in California. Both Sgr A* and the quasar calibrator 1924–292 were observed over three consecutive nights, and both sources were clearly detected on all baselines. For the first time, we are able to extract 1.3 mm VLBI interferometer phase information on Sgr A* through measurement of closure phase on the triangle of baselines. On the third night of observing, the correlated flux density of Sgr A* on all VLBI baselines increased relative to the first two nights, providing strong evidence for time-variable change on scales of a few Schwarzschild radii. These results suggest that future VLBI observations with greater sensitivity and additional baselines will play a valuable role in determining the structure of emission near the event horizon of Sgr A*.National Science Foundation (U.S.) (NSF Research Experiences for Undergraduates program)Academia Sinica (Beijing, China)Smithsonian Institution