  The first extragalactic X-ray binary, LMC X-1, was discovered in 1969. In the
1980s, its compact primary was established as the fourth dynamical black-hole
candidate. Recently, we published accurate values for the mass of the black
hole and the orbital inclination angle of the binary system. Building on these
results, we have analyzed 53 X-ray spectra obtained by RXTE and, using a
selected sample of 18 of these spectra, we have determined the dimensionless
spin parameter of the black hole to be a* = 0.92(-0.07,+0.05). This result
takes into account all sources of observational and model-parameter
uncertainties. The standard deviation around the mean value of a* for these 18
X-ray spectra, which were obtained over a span of several years, is only 0.02.
When we consider our complete sample of 53 RXTE spectra, we find a somewhat
higher value of the spin parameter and a larger standard deviation. Finally, we
show that our results based on RXTE data are confirmed by our analyses of
selected X-ray spectra obtained by the XMM-Newton, BeppoSAX and Ginga missions.
