Graphane functionalized with hydroxyl groups is shown to rapidly conduct protons under anhydrous conditions through a contiguous network of hydrogen bonds. Density functional theory calculations predict remarkably low barriers to diffusion of protons along a 1D chain of surface hydroxyls. Diffusion is controlled by the local rotation of hydroxyl groups, a mechanism that is very different from that found in 1D water wires in confined nanopores or in bulk water. The proton mean square displacement in the 1D chain was observed to follow Fickian diffusion rather than the expected single-file mobility. A charge analysis reveals that the charge on the proton is essentially equally shared by all hydrogens bound to oxygens, effectively delocalizing the proton.