  We report the results of a 16-month monitoring campaign using the Swift
satellite of PSR B0540-69, a young pulsar in the Large Magellanic Cloud. Phase
connection was maintained throughout the campaign so that a reliable ephemeris
could be determined, and the length of the campaign is adequate to accurately
determine the spin frequency and its first and second derivatives. The braking
index is 0.031 +/- 0.013 (90% confidence), a value much lower than previously
reported for B0540-69 and almost all other young pulsars. We use data from the
extensive monitoring campaign with RXTE to show that timing noise is unlikely
to significantly affect the measurement. This is the first measurement of the
braking index in the pulsar's recently discovered high spin-down state. We
discuss possible mechanisms for producing the low braking index.
