STUDIES of the kinematics of spiral galaxies suggest that dark (possibly non-baryonic) matter is a significant component of their total mass(1). It has been difficult to determine how far the dark-matter haloes of galaxies extend, however, because the presence of dark matter must be inferred from the motion of a luminous component such as atomic gas, which becomes hard to detect beyond several galactic radii. The recent suggestion(2) that the Lyman-a absorption lines seen in the spectra of distant quasars arise in the extended gaseous haloes of intervening galaxies offers an alternative means of probing their dark-matter content. Here we present observations of two low-luminosity spiral galaxies which have redshifts coincident with those of Lyman-a absorption lines in the spectra of quasars that are near the galaxies in the sky. The gas responsible for the absorption lines appears to take part in the rotation of the galaxies themselves, suggesting that the haloes of both galaxies are dominated by dark matter which remains a significant component of the haloes out to significantly greater radii than previously demonstrated.