We have studied an ultra-luminous X-ray source (ULX) in the dwarf galaxy NGC 5408 with a series of XMM-Newton observations, between 2001 July and 2003 January. We find that its X-ray spectrum is best fitted with a power law of photon index Gamma 2.6--2.9 and a thermal component with blackbody temperature kT_bb 0.12--0.14 keV. These spectral features, and the inferred luminosity 10^40 erg/s in the 0.3--12 keV band, are typical of bright ULXs in nearby dwarf galaxies. The blackbody plus power-law

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... model is a significantly better fit than either a simple power law or a broken power law (although the latter model is also acceptable at some epochs). Doppler-boosted emission from a relativistic jet is not required, although we cannot rule out this scenario. Our preliminary timing analysis shows flaring behaviour which we interpret as variability in the power-law component, on timescales of 10^2 s. The hard component is suppressed during the dips, while the soft thermal component is consistent with being constant. The power density spectrum is flat at low frequencies, has a break at f_b 2.5 mHz, and has a slope -1 at higher frequencies. A comparison with the power spectra of Cyg X-1 and of a sample of other BH candidates and AGN suggests a mass of 10^2 M_sun. It is also possible that the BH is at the upper end of the stellar-mass class (M 50 M_sun), in a phase of moderately super-Eddington accretion. The formation of such a massive BH via normal stellar evolution may have been favoured by the very metal-poor environment of NGC 5408.