We use a model of the Galactic fountain to simulate the neutral-hydrogen emission of the Milky Way Galaxy. The model was developed to account for data on external galaxies with sensitive H i data. For appropriate parameter values, the model reproduces well the H i emission observed at intermediate velocities. The optimal parameters imply that cool gas is ionized as it is blasted out of the disc, but becomes neutral when its vertical velocity has been reduced by ∼30 per cent. The parameters also imply that cooling of coronal gas in the wakes of fountain clouds transfers gas from the virial-temperature corona to the disc at ∼2 M⊙ yr−1. This rate agrees, to within the uncertainties, with the accretion rate required to sustain the Galaxy’s star formation without depleting the supply of interstellar gas. We predict the radial profile of accretion, which is an important input for models of Galactic chemical evolution. The parameter values required for the model to fit the Galaxy’s H i data are in excellent agreement with values estimated from external galaxies and hydrodynamical studies of cloud–corona interaction. Our model does not reproduce the observed H i emission at high velocities, consistent with high-velocity clouds being extragalactic in origin. If our model is correct, the structure of the Galaxy’s outer H i disc differs materially from that used previously to infer the distribution of dark matter on the Galaxy’s outskirts.