We present measurements of $e^+e^-$ production at midrapidity in Au$+$Au collisions at $sqrt{s_{_{NN}}}$ = 200 GeV. The invariant yield is studied within the PHENIX detector acceptance over a wide range of mass ($m_{ee} <$ 5 GeV/$c^2$) and pair transverse momentum ($p_T$ $<$ 5 GeV/$c$), for minimum bias and for five centrality classes. The ee yield is compared to the expectations from known sources. In the low-mass region ($m_{ee}=0.30$--0.76 GeV/$c^2$) there is an enhancement that increases with centrality and is distributed over the entire pair pt range measured. It is significantly smaller than previously reported by the PHENIX experiment and amounts to $2.3pm0.4({rm stat})pm0.4({rm syst})pm0.2^{rm model}$ or to $1.7pm0.3({rm stat})pm0.3({rm syst})pm0.2^{rm model}$ for minimum bias collisions when the open-heavy-flavor contribution is calculated with {sc pythia} or {sc mc@nlo}, respectively. The inclusive mass and $p_T$ distributions as well as the centrality dependence are well reproduced by model calculations where the enhancement mainly originates from the melting of the $rho$ meson resonance as the system approaches chiral symmetry restoration. In the intermediate-mass region ($m_{ee}$ = 1.2--2.8 GeV/$c^2$), the data hint at a significant contribution in addition to the yield from the semileptonic decays of heavy-flavor mesons.
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