An XMM-Newton 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 8 ^ { \prime \prime } 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 Einstein results , we derive an upper limit to the black-body 0.5–10 keV luminosity and emitting area of 1.8 \times 10 ^ { 32 } erg s ^ { -1 } and 1.3 \times 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 .