Aims. We analyse the magnetic field geometry of the magnetic DA white dwarf PG 1015+014 with our Zeeman tomography method.\ud Methods. This study is based on rotation-phase resolved optical flux and circular polarization spectra of PG 1015+014 obtained with FORS1\ud at the ESO VLT. Our tomographic code makes use of an extensive database of pre-computed Zeeman spectra. The general approach has been described in Papers I and II of this series.\ud Results. The surface field strength distributions for all rotational phases of PG 1015+014 are characterised by a strong peak at 70 MG. A separate peak at 80 MG is seen for about one third of the rotation cycle. Significant contributions to the Zeeman features arise from regions\ud with field strengths between 50 and 90 MG. We obtain equally good simultaneous fits to the observations, collected in five phase bins, for two diﬀerent field parametrizations: (i) a superposition of individually tilted and oﬀ-centred zonal multipole components; and (ii) a truncated multipole expansion up to degree l = 4 including all zonal and tesseral components. The magnetic fields generated by both parametrizations exhibit a similar global structure of the absolute surface field values, but diﬀer considerably in the topology of the field lines. An eﬀective photospheric temperature of T eﬀ = 10 000 ± 1000 K was found.\ud Conclusions. Remaining discrepancies between the observations and our best-fit models suggest that additional small-scale structure of the magnetic field exists which our field models are unable to cover due to the restricted number of free parameters.\u
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