The LMC pulsar PSR 0540-69, a young, rapidly rotating neutron star of dynamic age approximately 1000 yr, was observed with the Position Sensitive Proportional Counter (PSPC) onboard ROSAT for approximately 29,000 s in 1990 June, 1990 July, and 1991 February. The timing data are consistent with contemporaneous optical ephemerides suggesting that the optical and X-ray pulses are in phase. The characteristic broad sinusoidal pulse profile of PSR 0540-69 is observed at a pulsed fraction of approximately 15% measured against the pulsar plus the unresolved nebula. The spectral distribution of the pulsar + nebula counts is well described by a simple power law with a photon index of alpha = (2.4-1.85) and a neutral hydrogen column density of N(H) = (3.6-4.6) x 10(21) cm-2 at the 90% confidence level. The implied model luminosity in the 0.1-2.4 keV.band is 1.6 x 10(37) ergs s-1 which is approximately 11 % of the available rotational energy loss rate and similar to the value observed in the Crab nebula + pulsar. A positive correlation between the X-ray hardness and the pulse phase intensity implies an underlying spectrum for the pulsar which is harder than the nebula. A phase-resolved spectral fit to the pulsed component supports a harder spectrum for the pulsar and gives a power-law photon index of alpha(pulsed) = 1,3, The derived pulsed luminosity of 1.5 x 10(36) ergs s-1 is approximately 9% of the total X-ray luminosity.