We present a catalogue of 147 serendipitous X–ray sources selected to have hard spectra ( \alpha < 0.5 ) from a survey of 188 ROSAT fields .
Such sources must be the dominant contributors to the X-ray background at faint fluxes .
We have used Monte Carlo simulations to verify that our technique is very efficient at selecting hard sources : the survey has \geq 10 times as much effective area to hard sources as it has to soft sources above a 0.5 - 2 keV flux level of 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } .
The distribution of best fit spectral slopes of the hard sources suggests that a typical ROSAT hard source in our survey has a spectral slope \alpha \sim 0 .
The hard sources have a steep number flux relation ( dN / dS \propto S ^ { - \gamma } with a best fit value of \gamma = 2.72 \pm 0.12 ) and make up about 15 % of all 0.5 - 2 keV sources with S > 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } .
If their N ( S ) continues to fainter fluxes , the hard sources will comprise \sim 40 \% of sources with 5 \times 10 ^ { -15 } erg cm ^ { -2 } s ^ { -1 } < S < 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } .
The population of hard sources can therefore account for the harder average spectra of ROSAT sources with S < 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } .
They probably make a strong contribution to the X-ray background at faint fluxes and could be the solution to the X-ray background spectral paradox .