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We present an axi-symmetric model for the ultraviolet (UV)-to-submillimetre (submm) images of the nearly face-on spiral galaxy NGC 628. It was calculated using a radiative transfer (RT) code, accounting for the absorption and re-emission of starlight by dust in the interstellar medium of this galaxy. The code incorporates emission from Polycyclic Aromatic Hydrocarbons, anisotropic scattering, and stochastic heating of the grains. This is the second successful modelling of a face-on spiral galaxy with RT methods, whereby the large-scale geometry of stars and dust is self-consistently determined. The solution was obtained by fitting azimuthally averaged profiles in the UV, optical, and submm. The model predicts remarkably well all characteristics of the profiles, including the increase by a factor of 1.8 of the scale length of the infrared emissivity between 70 and 500 $\mu$m. We find that NGC 628 did not undergo an efficient inside-out disc growth, as predicted by semi-analytical hierarchical models for galaxy formation. We also find large amounts of dust grains at large radii, which could involve efficient transport mechanisms from the inner disc. Our results show that $71{{\ \rm per\ cent}}$ of the dust emission in NGC 628 is powered by the young stellar populations, with the old stellar populations from the bulge contributing $65{{\ \rm per\ cent}}$ to the heating of the dust in the central region (R < 0.5 kpc). The derived star formation rate is $\rm SFR=2.00\pm 0.15\, {\rm M}_{\odot }{\rm yr}^{-1}$.