If Fast Radio Bursts (FRBs) are truly at astronomical, in particular cosmological, distances, they represent one of the most exciting discoveries in astrophysics of the past decade. However, the distance to FRBs has, to date, been estimated purely from their excess dispersion, and has not been corroborated by any independent means. In this Letter, we discuss the possibility of detecting neutral hydrogen absorption against FRBs both from spiral arms within our own galaxy, or from intervening extragalactic H I clouds. In either case a firm lower limit on the distance to the FRB would be established. Absorption against galactic spiral arms may already be detectable for bright low-latitude bursts with existing facilities, and should certainly be so by the Square Kilometre Array (SKA). Absorption against extragalactic H I clouds, which would confirm the cosmological distances of FRBs, should also be detectable with the SKA, and maybe also Arecibo. Quantitatively, we estimate that SKA1-Mid should be able to detect H I absorption against about a few per cent of FRBs at a redshift z ˜ 1.