The interstellar matter in the direction of the supernova remnant G296.5+10.0 and the central X-ray source 1E 1207.4-5209

G296.5+10.0 is a high Galactic latitude supernova remnant (SNR), with a bilateral morphology in radio and X-rays. The compact X-ray source 1E 1207.4-5209, classified as a radio-quiet neutron star, is located very close to the remnant's center. We report on a survey of the H I distribution in a region of sky, 3.degrees 5 x 3.degrees 5, around the remnant, using the Australia Telescope Compact Array with a spatial resolution of about 3.'5 and a velocity resolution of 0.82 km s(-1). The H I distribution is quite smooth, with no obvious large-scale features that can explain the SNR bilateral morphology based on external factors. There are, however, three clouds that we do associate directly with the remnant. Optical filaments outline two smaller features, which appear to have been overtaken by the shock front and are cooling radiatively. Also, a high-velocity cloud gives a lower limit of similar to 35 km s(-1) to the expansion velocity of the shock into the H I gas and a lower limit of similar to 2 x 10(49) ergs for the kinetic energy injected by the supernova explosion into the surrounding interstellar medium. We estimate a distance to the SNR of d = 2.1(-0.8)(+1.8) kpc and a total mass of at least 1900 M . of associated H I gas. A hole in the H I column density is observed at the same position of the compact X-ray source 1E 1207.4-5209. This H I hole is at the same radial velocity as G296.5+10.0. We argue that this result constitutes concrete evidence that 1E 1207.4-5209 and G296.5+10.0 are physically associated.