  The PHENIX experiment at RHIC has measured the centrality dependence of the
direct photon yield from Au$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV down
to $p_T=0.4$ GeV/$c$. Photons are detected via photon conversions to $e^+e^-$
pairs and an improved technique is applied that minimizes the systematic
uncertainties that usually limit direct photon measurements, in particular at
low $p_T$. We find an excess of direct photons above the $N_{\rm coll}$-scaled
yield measured in $p$$+$$p$ collisions. This excess yield is well described by
an exponential distribution with an inverse slope of about 240 MeV/$c$ in the
$p_T$ range from 0.6--2.0 GeV/$c$. While the shape of the $p_T$ distribution is
independent of centrality within the experimental uncertainties, the yield
increases rapidly with increasing centrality, scaling approximately with
$N_{\rm part}^\alpha$, where $\alpha=1.48{\pm}0.08({\rm stat}){\pm}0.04({\rm
syst})$.
