Extensive Ginga observations of PSR 0540-69, the Crab-like 50 ms pulsar in the Large Magellanic Cloud, have been obtained as a side benefit of a pulsar search project for SN 1987A. Through a coherent pulse timing analysis of data from 46 separate days between 1987 July and 1988 October, we have obtained precise values for the pulse frequency and its first and second derivatives. From these values, a braking index of n = 2.02±0.01 is obtained for PSR 0540-69. This is the first accurate measurement of a pulsar braking index from X-ray observations and the third overall. The braking index for PSR 0540-69 is much smaller than those previously determined for the Crab pulsar (n = 2.51) and PSR 1509-58 (n = 2.83).