We report on the discovery of stellar occultations, observed with Kepler, which recur periodically at 15.685 hr intervals, but which vary in depth from a maximum of 1.3% to a minimum that can be less than 0.2%. The star that is apparently being occulted is KIC 12557548, a V = 16 mag K dwarf with T-eff,T-s similar or equal to 4400 K. The out-of-occultation behavior shows no evidence for ellipsoidal light variations, indicating that the mass of the orbiting object is less than similar to 3 M-J (for an orbital period of 15.7 hr). Because the eclipse depths are highly variable, they cannot be due solely to transits of a single planet with a fixed size. We discuss but dismiss a scenario involving a binary giant planet whose mutual orbit plane precesses, bringing one of the planets into and out of a grazing transit. This scenario seems ruled out by the dynamical instability that would result from such a configuration. We also briefly consider an eclipsing binary, possibly containing an accretion disk, that either orbits KIC 12557548 in a hierarchical triple configuration or is nearby on the sky, but we find such a scenario inadequate to reproduce the observations. The much more likely explanation-but one which still requires more quantitative development-involves macroscopic particles escaping the atmosphere of a slowly disintegrating planet not much larger than Mercury in size. The particles could take the form of micron-sized pyroxene or aluminum oxide dust grains. The planetary surface is hot enough to sublimate and create a high-Z atmosphere; this atmosphere may be loaded with dust via cloud condensation or explosive volcanism. Atmospheric gas escapes the planet via a Parker-type thermal wind, dragging dust grains with it. We infer a mass-loss rate from the observations of order 1 M-circle plus Gyr(-1), with a dust-to-gas ratio possibly of order unity. For our fiducial 0.1 M-circle plus planet (twice the mass of Mercury), the evaporation timescale may be similar to 0.2 Gyr. Smaller mass planets are disfavored because they evaporate still more quickly, as are larger mass planets because they have surface gravities too strong to sustain outflows with the requisite mass-loss rates. The occultation profile evinces an ingress-egress asymmetry that could reflect a comet-like dust tail trailing the planet; we present simulations of such a tail.