We present the discovery of a very faint stellar system, SMASH 1, that is potentially a satellite of the Large Magellanic Cloud. Found within the Survey of the MAgellanic Stellar History (SMASH), SMASH 1 is a compact (r(h) 9.1(-3.4)(+5.9)pc) and very low luminosity (M-V = -1.0 +/- 0.9, L-V = 10(2.3 +/- 0.4) L-circle dot) stellar system that is revealed by its sparsely populated main sequence and a handful of red giant branch candidate member stars. The photometric properties of these stars are compatible with a metal-poor ([Fe/H] = -2.2) and old (13 Gyr) isochrone located at a distance modulus of similar to 18.8, i.e., a distance of similar to 57 kpc. Situated at 11 degrees.3 from the LMC in projection, its three-dimensional distance from the Cloud is similar to 13 kpc, consistent with a connection to the LMC, whose tidal radius is at least 16 kpc. Although the nature of SMASH 1 remains uncertain, its compactness favors it being a stellar cluster and hence dark-matter free. If this is the case, its dynamical tidal radius is only less than or similar to 19 pc at this distance from the LMC, and smaller than the system's extent on the sky. Its low luminosity and apparent high ellipticity (epsilon = 0.62(-0.21)(+0.17)) with its major axis pointing toward the LMC may well be the tell-tale sign of its imminent tidal demise.