  We report on a new estimate of the QSO X-ray luminosity function and its
evolution with redshift based on a sample of 107 QSOs detected at faint X-ray
fluxes, $S{\rm(0.5-2\,keV)}>4\times10^{-15}\,$\ergcms , with the {\it ROSAT}
X-ray satellite. For $q_0=0.5$, the X-ray evolution of QSOs in this sample is
consistent with strong luminosity evolution, $L_{\rm X} ^*(z) \propto
(1+z)^{3.25\pm0.1}$, at low redshifts ($z<1.60$) and a constant comoving space
density at higher redshifts. The derived rate of evolution at low redshifts is
thus significantly higher than that obtained previously for the {\it Einstein}
Extended Medium Sensitivity Survey (EMSS). Indeed, most luminosity evolution
models provide a very poor fit (rejected at the 99 per cent confidence level)
when applied to the combined EMSS and {\it ROSAT} samples, although a
polynomial evolution model, $L_{\rm X} ^*(z) \propto 10^{(1.14\,z
-0.23\,z^2)}$, provides an adequate fit for $q_0=0$. For $q_0=0.5$, a simple
power-law luminosity evolution model with a redshift cut-off ($L_{\rm X} ^*(z)
\propto (1+z)^{2.51\pm0.1}$,
