We present the results of mid-infrared nulling interferometric observations of the main-sequence star alpha Lyr (Vega) using the 6.5 m MMT with its adaptive secondary mirror. From the observations at 10.6 microns, we find that there is no resolved emission from the circumstellar environment (at separations greater than 0.8 AU) above 2.1% (3 sigma limit) of the level of the stellar photospheric emission. Thus, we are able to place an upper limit on the density of dust in the inner system of 650 times that of our own solar systems zodiacal cloud. This limit is roughly 2.8 times better than those determined with photometric excess observations such as those by IRAS. Comparison with far-infrared observations by IRAS shows that the density of warm dust in the inner system (< 30 AU) is significantly lower than cold dust at larger separations. We consider two scenarios for grain removal, the sublimation of ice grains and the presence of a planetary mass sweeper. We find that if sublimation of ice grains is the only removal process, a large fraction (> 80%) of the material in the outer system is ice.