  We present the first good evidence for exocomet transits of a host star in
continuum light in data from the Kepler mission. The Kepler star in question,
KIC 3542116, is of spectral type F2V and is quite bright at K_p = 10. The
transits have a distinct asymmetric shape with a steeper ingress and slower
egress that can be ascribed to objects with a trailing dust tail passing over
the stellar disk. There are three deeper transits with depths of ~0.1% that
last for about a day, and three that are several times more shallow and of
shorter duration. The transits were found via an exhaustive visual search of
the entire Kepler photometric data set, which we describe in some detail. We
review the methods we use to validate the Kepler data showing the comet
transits, and rule out instrumental artefacts as sources of the signals. We fit
the transits with a simple dust-tail model, and find that a transverse comet
speed of ~35-50 km/s and a minimum amount of dust present in the tail of ~10^16
g are required to explain the larger transits. For a dust replenishment time of
~10 days, and a comet lifetime of only ~300 days, this implies a total cometary
mass of > 3 x 10^17 g, or about the mass of Halley's comet. We also discuss the
number of comets and orbital geometry that would be necessary to explain the
six transits detected over the four years of Kepler prime-field observations.
Finally, we also report the discovery of a single comet-shaped transit in KIC
11084727 with very similar transit and host-star properties.
