  The {\it Transiting Exoplanet Survey Satellite} (TESS) searches for planets
transiting bright and nearby stars using high-cadence, large-scale photometric
observations. Full Frame Images provided by the TESS mission include large
number of serendipitously observed main-belt asteroids. Due to the cadence of
the published Full Frame Images we are sensitive to periods as long as of order
tens of days, a region of phase space that is generally not accessible through
traditional observing. This work represents a much less biased measurement of
the period distribution in this period range. We have derived rotation periods
for 300~main-belt asteroids and have partial lightcurves for a further 7277
asteroids, including 43 with periods $P > 100$ h; this large number of slow
rotators is predicted by theory. Of these slow rotators we find none requiring
significant internal strength to resist rotational reshaping. We find our
derived rotation periods to be in excellent agreement with results in the
Lightcurve Database for the 55~targets that overlap. Over the nominal two-year
lifetime of the mission, we expect the detection of around 85,000 unique
asteroids with rotation period solutions for around 6000 asteroids. We project
that the systematic analysis of the entire TESS data set will increase the
number of known slow-rotating asteroids (period > 100~h) by a factor of 10.
Comparing our new period determinations with previous measurements in the
literature, we find that the rotation period of asteroid (2320) Blarney has
decreased by at least 20\% over the past decade, potentially due to surface
activity or subcatastrophic collisions.
