  We report here on the discovery of stellar occultations, observed with
Kepler, that recur periodically at 15.685 hour 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 K dwarf with T_eff = 4400
K and V = 16. 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. We
also briefly consider an eclipsing binary, 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. We come down in favor of an
explanation that involves macroscopic particles escaping the atmosphere of a
slowly disintegrating planet not much larger than Mercury. 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_E/Gyr, with a dust-to-gas ratio possibly of order unity. For our
fiducial 0.1 M_E planet, the evaporation timescale may be ~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.
