We report on more than 7 years of monitoring of PSR J0537-6910, the 16 ms pulsar in the Large Magellanic Cloud, using data acquired with the RXTE. During this campaign the pulsar experienced 23 sudden increases in frequency (``glitches) amounting to a total gain of over six ppm of rotation frequency superposed on its gradual spindown of d(nu)/d(t) = -2e-10 Hz/s. The time interval from one glitch to the next obeys a strong linear correlation to the amplitude of the first glitch, with a mean slope of about 400 days ppm (6.5 days per uHz), such that these intervals can be predicted to within a few days, an accuracy which has never before been seen in any other pulsar. There appears to be an upper limit of ~40 uHz for the size of glitches in_all_ pulsars, with the 1999 April glitch of J0537 as the largest so far. The change in the spindown of J0537 across the glitches, Delta(d(nu)/d(t)), appears to have the same hard lower limit of -1.5e-13 Hz/s, as, again, that observed in all other pulsars. The spindown continues to increase in the long term, d(d(nu)/d(t))/d(t) = -1e-21 Hz/s/s, and thus the timing age of J0537 (-0.5 nu d(nu)/d(t)) continues to decrease at a rate of nearly one year every year, consistent with movement of its magnetic moment away from its rotational axis by one radian every 10,000 years, or about one meter per year. J0537 was likely to have been born as a nearly-aligned rotator spinning at 75-80 Hz, with a |d(nu)/d(t)| considerably smaller than its current value of 2e-10 Hz/s. The pulse profile of J0537 consists of a single pulse which is found to be flat at its peak for at least 0.02 cycles.
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