We present the first mid-infrared detection of a source coincident with the black hole candidate Sgr A* in a deep image of the inner parsec of the Galaxy. These data were obtained with SpectroCam-10 at the 200-inch Hale telescope. The source was detected at 8.7 micron on two separate occasions using different observing techniques. The spatial resolution is ~ 0.''7 and the noise level is 1.6 mJy beam^-1 in the combined mosaic near Sgr A*. A source at Sgr A* is weakly apparent in the raw data, but is difficult to identify due to strong diffuse dust emission with a mean flux density of 500 mJy arcsec^-2. Several deconvolution techniques were applied to achieve the diffraction limit of 0.''44, all of which revealed a peak on a narrow NW-SE ridge of emission. The peak is coincident with Sgr A* within the +/-0.''3 uncertainty in locating Sgr A* relative to IRS 7. We estimate the flux density of the Sgr A* source above the ridge to be 25 +/- 5 mJy. The extinction-corrected 8.7 micron flux density of ~ 100 mJy is significantly greater than what is predicted by dustless accretion disk models. The excess emission is best explained by warm dust which may be associated with Sgr A* or heated by local stellar sources.