Direct photon production at mid-rapidity in Pb–Pb collisions at was studied in the transverse momentum range k 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 k 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 k to view the MathML source">pT≳5 GeV/c. Direct photon spectra down to k 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 k to view the MathML source">0.9<pT<2.1 GeV/c is 2.6σ for the 0–20% class. The spectrum in this k to view the MathML source">pT range and centrality class can be described by an exponential with an inverse slope parameter of k 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.