Terrestrial exospheric atomic hydrogen (H) resonantly scatters solar Lyman-α (121.567 nm) radiation, observed as the glow of the H-geocorona. The Two Wide-angle Imaging Neutral-atom Spectrometers (TWINS) satellites are equiped with two Lyman-α line-of-sight Detectors (LADs) each. Since during the past solar minimum conditions the relevant solar control parameters practically did not vary, we are using LAD data between June and September 2008 to create a time averaged hydrogen geocorona model representative for these solar minimum conditions. In this averaged model we assume that the H-geocorona is longitudinally symmetric with respect to the earth-sun line. We find a 3-dimensional H-density distribution in the range from 3 to 8 earth radii which with some caution can also be extrapolated to larger distances. For lower geocentric distances than 3 earth radii a best fitting r-dependent Chamberlain (1963)-like model is adapted. Main findings are larger than conventionally expected H-densities at heights above 5 R_E and a pronounced day-to-night side H-density asymmetry. The H-geocorona presented here should serve as a reference H-atmosphere for the earth during solar minimum conditions.