  The alignment of Saturn's magnetic pole with its rotation axis precludes the
use of magnetic field measurements to determine its rotation period. The period
was previously determined from radio measurements by the Voyager spacecraft to
be 10h 39m 22.4s. When the Cassini spacecraft measured a period of 10h 47m 6s,
which was additionally found to change between sequential measurements, it
became clear that the radio period could not be used to determine the bulk
planetary rotation period. Estimates based upon Saturn's measured wind fields
have increased the uncertainty even more, giving numbers smaller than the
Voyager rotation period, and at present Saturn's rotation period is thought to
be between 10h 32m and 10h 47m, which is unsatisfactory for such a fundamental
property. Here we report a period of 10h 32m 45s +- 46s, based upon an
optimization approach using Saturn's measured gravitational field and limits on
the observed shape and possible internal density profiles. Moreover, even when
solely using the constraints from its gravitational field, the rotation period
can be inferred with a precision of several minutes. To validate our method, we
applied the same procedure to Jupiter and correctly recovered its well-known
rotation period.
