  We report systematic measurements of dielectron ($e^{\pm}e^{\pm}$)
invariant-mass $M_{ee}$ spectra at mid-rapidity in Au+Au collisions at
$\sqrt{s_{NN}}$ = 27, 39, and 62.4 GeV taken with the STAR detector at the
Relativistic Heavy Ion Collider. For all energies studied, a significant excess
yield of dielectrons is observed in the low-mass region (0.40$ < M_{ee} < 0.75$
MeV/$c^2$) compared to hadronic cocktail simulations at freeze-out. Models that
include an in-medium broadening of the $\rho$-meson spectral function
consistently describe the observed excess. In addition, we report
acceptance-corrected dielectron-excess spectra for Au+Au collisions at
mid-rapidity ($\left|y_{ee}\right|$ $<$ 1) in the 0$-$80% centrality bin for
each collision energy. The integrated excess yields for $0.4 < M_{ee} < 0.75\
\textrm{GeV}/c^{2}$, normalized by the charged particle multiplicity at
mid-rapidity, are compared with previously published measurements for Au+Au at
$\sqrt{s_{NN}}$ = 19.6 and 200 GeV. The normalized excess yields in the
low-mass region show no significant collision energy dependence. The data,
however, are consistent with model calculations that demonstrate a modest
energy dependence.
