We present the diffuse X-ray emission identified in Chandra observations of the young, massive Galactic star cluster Westerlund 1. After removing point-like X-ray sources down to a completeness limit of 2e31 erg/s, we identify 3e34 erg/s (2--8 keV) of diffuse emission. The spatial distribution of the emission can be described as a slightly-elliptical Lorentzian core with a half-width half-maximum along the major axis of 25+/-1, similar to the distribution of point sources in the cluster, plus a 5 halo of extended emission. The spectrum of the diffuse emission is dominated by a hard continuum component that can be described as a kT>3 keV thermal plasma that has a low iron abundance (<0.3 solar), or as non-thermal emission that could be stellar light that is inverse-Compton scattered by MeV electrons. Only 5% of the flux is produced by a kT=0.7 keV plasma. The low luminosity of the thermal emission and the lack of a 6.7 keV iron line suggests that <40,000 unresolved stars with masses between 0.3 and 2 Msun are present in the cluster. Moreover, the flux in the diffuse emission is a factor of two lower than would be expected from a supersonically-expanding cluster wind, and there is no evidence for thermal remnants produced by supernovae. Less than 1e-5 of the mechanical luminosity of the cluster is dissipated as 2--8 keV X-rays, leaving a large amount of energy that either is radiated at other wavelengths, is dissipated beyond the bounds of our image, or escapes into the intergalactic medium.
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