  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.
