  <b>Aims:</b> We analyse the magnetic field geometry of the magnetic DA white
dwarf PG 1015+014 with our Zeeman tomography method.<br> <b>Methods:</b> This
study is based on rotation-phase resolved optical flux and circular
polarization spectra of PG 1015+014 obtained with FORS1 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.<br> <b>Results:</b> 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 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<sub>eff</sub> = 10000
&plusmn; 1000 K was found.<br> <b>Conclusions:</b> 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.<br>
