We present ASCA GIS observations (total exposure 100-200 ksec) of three fields which form part of our deep ROSAT survey. We detect 26 sources down to a limiting flux (2-10 keV) of 5x10^{-14} erg cm-2 s-1. Sources down to this flux level contribute 30 per cent of the 2-10 keV X-ray background. The number-count distribution, logN-logS, is a factor of three above the ROSAT counts, assuming a photon spectral index of 2 for the ROSAT sources. This suggests the presence at hard energies of a population other than the broad-line AGN which contribute to the ROSAT counts. This is supported by spectroscopic observations that show a large fraction of sources that are not obvious broad-line AGN. The average 1-10 keV spectral index of these sources is flat (0.92+-0.16), significantly different than that of the broad-line AGN (1.78+-0.16). Although some of the Narrow Emission Line Galaxies which are detected with ROSAT are also detected here, the nature of the flat spectrum sources remains as yet unclear. ; Comment: 8 pages, Latex. To appear in MNRAS