  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.
