We present a rigorous theoretical analysis of the ALICE measurement of low-p_T direct-photon production in central lead-lead collisions at the LHC with a centre-of-mass energy of sqrt{s_{NN}}=2.76 TeV. Using NLO QCD, we compute the relative contributions to prompt-photon production from different initial and final states and the theoretical uncertainties coming from independent variations of the renormalisation and factorisation scales, the nuclear parton densities and the fragmentation functions. Based on different fits to the unsubtracted and prompt-photon subtracted ALICE data, we consistently find T = 304 pm 58 MeV and 309 pm 64 MeV for the effective temperature of the quark-gluon plasma (or hot medium) at p_T in [0.8;2.2] GeV and p_T in [1.5;3.5] GeV as well as a power-law (p_T^{-4}) behavior for p_T > 4 GeV as predicted by QCD hard scattering.
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