Aims. The helium-peculiar star a Cen exhibits interesting line profile variations of elements such as iron, nitrogen and oxygen in addition to its well-known extreme helium variability. The objective of this paper is to use new high signal-to-noise, high-resolution spectra to perform a quantitative measurement of the helium, iron, nitrogen and oxygen abundances of the star and determine the relation of the concentrations of the heavier elements on the surface of the star to the helium concentration and perhaps to the magnetic field orientation. Methods. Doppler images have been created for the elements helium, iron, nitrogen and oxygen using the programs described in earlier papers by Rice and others. An alternative surface abundance mapping code has been used to model the helium line variations after our Doppler imaging of certain individual helium lines produced mediocre results. Results. Doppler imaging of the helium abundance of a Cen confirms the long-known existence of helium-rich and helium-poor hemispheres on the star and we measure a difference of more than two orders of magnitude in helium abundance from one side of the star to the other. Helium is overabundant by a factor of about 5 over much of the helium-rich hemisphere. Of particular note is our discovery that the helium-poor hemisphere has a very high abundance of 3 He, approximately equal to the 4 He abundance. a Cen is therefore a new member of the small group of helium-3 stars and the first well-established magnetic member of the class. For the three metals investigated here, there are two strong concentrations of abundance near the equator at longitude roughly 135 • consistent with the positive magnetic maximum and two somewhat weaker concentrations of abundance near longitude 315 • on the equator near where the helium concentration is centered and roughly where the negative peak of the magnetic field would be found. Another strong concentration is found near the equator at about longitude 45 • and this is not explainable in terms of any simple symmetry with the helium abundance or the apparent magnetic field main polar locations.