Aims: We present a study of the diffuse X-ray emission in the halo and the disc of the starburst galaxy NGC 253. Methods: After removing point-like sources, we analysed XMM-Newton images, hardness ratio maps and spectra from several regions in the halo and the disc. We introduce a method to produce vignetting corrected images from the EPIC pn data, and we developed a procedure that allows a correct background treatment for low surface brightness spectra, using a local background, together with closed filter observations. Results: Most of the emission from the halo is at energies below 1 keV. In the disc, also emission at higher energies is present. The extent of the diffuse emission along the major axis of the disc is 13.6 kpc. The halo resembles a horn structure and reaches out to ~9 kpc perpendicular to the disc. Disc regions that cover star forming regions, like spiral arms, show harder spectra than regions with lower star forming activity. Models for spectral fits of the disc regions need at least three components: two thermal plasmas with solar abundances plus a power law and galactic foreground absorption. Temperatures are between 0.1 and 0.3 keV and between 0.3 and 0.9 keV for the soft and the hard component, respectively. The power law component may indicate an unresolved contribution from X-ray binaries in the disc. The halo emission is not uniform, neither spatially nor spectrally. The southeastern halo is softer than the northwestern halo. To model the spectra in the halo, we needed two thermal plasmas with solar abundances plus galactic foreground absorption. Temperatures are around 0.1 and 0.3 keV. A comparison between X-ray and UV emission shows that both originate from the same regions.
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