A numerical solution is found for the accretion of an intergalactic gas by a stationary, realistic model of the Galaxy in which the flow is unimpeded by the interstellar pressure of the disc, in which case the accretion rate is a maximum. The calculation was performed for a range of initial temperatures for the intergalactic gas and for three values of its specific heat ratio including both adiabatic and isothermal cases. The accretion rates derived are quite large being, in every case, greater than 5.6 M yr_1 if the initial density of the gas is 3 x 10-4 cm- 3. If the angular momentum of the incoming gas were zero, such high accretion rates would cause velocity differences between gas and stars in the disc which could only marginally be consistent with observation for certain initial temperatures. The observed pressure in the disc matches the pressure of the accreting gas, but it would also match the pressure of flows having lower accretion rates. In view of the angular momentum difficulty it is possible that the accretion flow is in fact impeded by the pressure in the disc. Soft X-radiation from the inflowing gas having initial density ~ 3 x 10-4 cm-3 and initial temperature greater than ~ 2.5 x 105 K would constitute an appreciable fraction of the observed soft X-ray flux and would provide a more local source for the extragalactic component of Davidsen's model for the soft X-ray flux. This would make the model consistent with the observed lack of absorption by neutral hydrogen in the Small Magellanic Cloud. The variation of heavy element abundance with epoch, if the accreted gas contains no heavy elements, is consistent with observation for moderate accretion rates. When accretion is present the unastrated fraction of the interstellar medium is increased, which, assuming the observed deuterium abundance is made wholly in the ' big-bang ', would imply an increase in the estimate for the mean density of the Universe. © Royal Astronomical Society.