We derive the projected surface mass distribution ∑M for spherically symmetric mass distributions having an arbitrary rotation curve. For a galaxy with a flat rotation curve and an interstellar medium (ISM) disc having a constant Toomre stability parameter, Q, the ISM surface mass density ∑g and ∑M both fall off as R-1. We use published data on a sample of 20 well-studied galaxies to show that ISM discs do maintain a constant Q over radii usually encompassing more than 50 per cent of the HI mass. The power-law slope in ∑g covers a range of exponents and is well correlated with the slope in the epicyclic frequency. This implies that the ISM disc is responding to the potential, and hence that secular evolution is important for setting the structure of ISM discs. We show that the gas-to-total mass ratio should be anticorrelated with the maximum rotational velocity, and that the sample falls on the expected relationship. A very steep fall-off in ∑g is required at the outermost radii to keep the mass and angular momentum content finite for typical rotation curve shapes, and is observed. The observation that HI traces dark matter over a significant range of radii in galaxies is thus due to the discs stabilizing themselves in a normal dark matter dominated potential. This explanation is consistent with the cold dark matter paradigm. © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.