We use Spitzer data to infer that the small infrared excess of V819 Tau, a weak-lined T Tauri star in Taurus, is real and not attributable to a "companion" 10 arcsec to the south. We do not confirm the mid-infrared excess in HBC 427 and V410 X-ray 3, which are also non-accreting T Tauri stars in the same region; instead, for the former object, the excess arises from a red companion 9 arcsec to the east. A single-temperature blackbody fit to the continuum excess of V819 Tau implies a dust temperature of 143 K; however, a better fit is achieved when the weak 10 and 20 micron silicate emission features are also included. We infer a disk of sub-micron silicate grains between about 1 AU and several 100 AU with a constant surface density distribution. The mid-infrared excess of V819 Tau can be successfully modeled with dust composed mostly of small amorphous olivine grains at a temperature of 85 K, and most of the excess emission is optically thin. The disk could still be primordial, but gas-poor and therefore short-lived, or already at the debris disk stage, which would make it one of the youngest debris disk systems known.