The PHENIX experiment at RHIC has measured the centrality dependence of the direct photon yield from Au + Au collisions at root s(NN) = 200 GeV down to pT = 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 pT. We find an excess of direct photons above the N-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 pT range 0.6-2.0 GeV/c. While the shape of the pT distribution is independent of centrality within the experimental uncertainties, the yield increases rapidly with increasing centrality, scaling approximately with N-part(alpha), where alpha = 1.38 +/- 0.03(stat) +/- 0.07(syst).