Many spiral galaxies have stellar bars-an elongated region near the centre from which spiral arms emerge. Gas and stars in barred galaxies move on highly non-circular orbits. Models(1-5) predict that the gas streamlines are strongly deflected along shock fronts in the bar region, and that the gas behind the shock is compressed. Dust lanes form in dense gas regions and it is believed that gas flows inward along these lanes to fuel bursts of star formation in a ring of dense molecular gas near the centre of the galaxy(6,7). This inflow is difficult to measure observationally(8-10). Magnetic fields are known to pervade the interstellar gas in all spiral galaxies(11), but the relationship between such fields and the gas flow in barred galaxies has not hitherto been investigated. Here we report high-resolution radio observations of the magnetic fields in the barred galaxy NGC1097. We find a regular magnetic field in the bar and in the circumnuclear ring. The field in the bar is well aligned with the theoretical gas streamlines, and so appears to be a good tracer of the gas how The magnetic stress in the ring can drive mass inward at the rate needed to fuel the active nucleus of this galaxy.