Direct photon production at mid-rapidity in Pb–Pb collisions at md5=1c8185232f9746b818569f8c750d7a2c"> was studied in the transverse momentum range md5=0013d3c9759dbd106f0bce3f81d2dae8" title="Click to view the MathML source">0.9<pT<14 GeV/c. Photons were detected with the highly segmented electromagnetic calorimeter PHOS and via conversions in the ALICE detector material with the md5=1166737fefdecce3d6290006e66f4e08" title="Click to view the MathML source">e+e− pair reconstructed in the central tracking system. The results of the two methods were combined and direct photon spectra were measured for the 0–20%, 20–40%, and 40–80% centrality classes. For all three classes, agreement was found with perturbative QCD calculations for md5=f876eeae8231e04158b538b59eb9bcfd" title="Click to view the MathML source">pT≳5 GeV/c. Direct photon spectra down to md5=145471da8223133907c91fcc1011f6da" title="Click to view the MathML source">pT≈1 GeV/c could be extracted for the 20–40% and 0–20% centrality classes. The significance of the direct photon signal for md5=4067fc100418e5d9dc8ef576fcc34c21" title="Click to view the MathML source">0.9<pT<2.1 GeV/c is 2.6σ for the 0–20% class. The spectrum in this md5=aa58cb03581d230341ba1ff810fef8b0" title="Click to view the MathML source">pT range and centrality class can be described by an exponential with an inverse slope parameter of md5=2e8066a435697eed598476449755381f" title="Click to view the MathML source">(297±12stat±41syst) MeV. State-of-the-art models for photon production in heavy-ion collisions agree with the data within uncertainties.