13 pages, 6 figures.-- Accepted for publication in MNRAS.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 > 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. 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 ~0. The hard sources have a steep number flux relation (dN/dS propto S^-gamma with a best fit value of gamma = 2.72 +- 0.12) and make up about 15% of all 0.5 - 2 keV sources with S > 10^-14 erg/cm^2/s. If their N(S) continues to fainter fluxes, the hard sources will comprise ~ 40% of sources with 5 10^-15 < S < 10^-14. The population of hard sources can therefore account for the harder average spectra of ROSAT sources with S < 10^-14. 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.FJC thanks the DGES for partial financial support, under project PB95-0122.Peer reviewed