The production of e+ e- pairs for m(e+ e-)<0.3 GeV/c2 and 1<p(T)<5 GeV/c is measured in p+p and Au+Au collisions at square root(S(NN))=200 GeV. An enhanced yield above hadronic sources is observed. Treating the excess as photon internal conversions, the invariant yield of direct photons is deduced. In central Au+Au collisions, the excess of the direct photon yield over p+p is exponential in transverse momentum, with an inverse slope T=221+/-19(stat)+/-19(syst) MeV. Hydrodynamical models with initial temperatures ranging from T(init) approximately 300-600 MeV at times of approximately 0.6-0.15 fm/c after the collision are in qualitative agreement with the data. Lattice QCD predicts a phase transition to quark gluon plasma at approximately 170 MeV.