An xmm observation of the plerionic supernova remnant 3C58 has allowed us to study the X-ray nebula with unprecedented detail. A spatially resolved spectral analysis with a resolution of 8arcsec has yielded a precise determination of the relation between the spectral index and the distance from the center. We do not see any evidence for bright thermal emission from the central core. In contrast with previous ASCA and {em Einstein} results, we derive an upper limit to the black-body 0.5-10 keV luminosity and emitting area of $1.8times 10^{32}$ ergsec and $1.3times 10^{10}$ cm$^2$, respectively, ruling out emission from the hot surface of the putative neutron star and also excluding the outer-gap model for hot polar caps. We have performed for the first time a spectral analysis of the outer regions of the X-ray nebula, where most of the emission is still non-thermal, but where the addition of a soft (kT=0.2-0.3 keV) optically thin plasma component is required to fit the spectrum at $E<1$ keV. This component provides 6% of the whole remnant observed flux in the 0.5-10.0 keV band. We show that a Sedov interpretation is incompatible with the SN1181-3C58 association, unless there is a strong deviation from electron-ion energy equipartition, and that an origin of this thermal emission in terms of the expansion of the nebula into the ejecta core nicely fits all the radio and X-ray observations.
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