The electronic structure of a single monatomic gold wire has been calculated with density-functional calculations taking into account relativistic effects. For stretched structures in the experimentally accessible range the conduction band is exactly half-filled, whereas the band structures are more complex for the optimized structure. Moreover, for these structures the system can lower its total energy by letting the bond lengths alternate leading to more or less separated dimers with bond lengths of about 2.5 Angstrom, largely independent of the stretching. Finally, finite Au-2, Au-4, and Au-6 chains possess electronic properties very similar to those of the infinite chain.