ABSTRACT 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 n 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 Rossi X-ray Timing Explorer 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.