  We show that the gas giant exoplanet HD 189733b is less oblate than Saturn,
based on Spitzer Space Telescope photometry of seven transits. The observable
manifestations of oblatenesswould have been slight anomalies during the ingress
and egress phases, as well as variations in the transit depth due to spin
precession. Our nondetection of these effects gives the first empirical
constraints on the shape of an exoplanet. The results are consistent with the
theoretical expectation that the planetary rotation period and orbital period
are synchronized, in which case the oblateness would be an order of magnitude
smaller than our upper limits. Conversely, if HD 189733b is assumed to be in a
synchronous, zero-obliquity state, then the data give an upper bound on the
quadrupole moment of the planet (J2 < 0.068 with 95% confidence) that is too
weak to constrain the interior structure of the planet. An Appendix describes a
fast algorithm for computing the transit light curve of an oblate planet, which
was necessary for our analysis.
