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We report on measurements of dielectron (e⁺e⁻) production in Au+Au collisions at a center-of-mass energy of 200 GeV per nucleon-nucleon pair using the STAR detector at RHIC. Systematic measurements of the dielectron yield as a function of transverse momentum (pT) and collision centrality show an enhancement compared to a cocktail simulation of hadronic sources in the low invariant-mass region (Mee < 1GeV/c2). This enhancement cannot be reproduced by the ρ-meson vacuum spectral function. In minimum-bias collisions, in the invariant-mass range of 0.30 – 0.76GeV/c², integrated over the full pT acceptance, the enhancement factor is 1.76±0.06(stat.)±0.26(sys.)±0.29(cocktail). The enhancement factor exhibits weak centrality and pT dependence in STAR's accessible kinematic regions, while the excess yield in this invariant-mass region as a function of the number of participating nucleons follows a power-law shape with a power of 1.44±0.10. Models that assume an in-medium broadening of the ρ-meson spectral function consistently describe the observed excess in these measurements. In addition, we report on measurements of ω- and Φ-meson production through their e⁺e⁻ decay channel. These measurements show good agreement with Tsallis blast-wave model predictions, as well as, in the case of the Φ meson, results through its K⁺K⁻ decay channel. In the intermediate invariant-mass region (1.1 < Mee < 3GeV/c²), we investigate the spectral shapes from different collision centralities. Physics implications for possible in-medium modification of charmed hadron production and other physics sources are discussed.