  We present a systematic 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 next-to-leading
order of perturbative 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 an exponential,
possibly thermal, photon spectrum from the quark-gluon plasma (or hot medium)
with slope $T=304\pm 58$ MeV and $309\pm64$ MeV 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.
