  Using $(27.12\pm 0.14)\times10^8$ $\psi(3686)$ decays and data samples of
$e^+e^-$ collisions with $\sqrt{s}$ from 4.130 to 4.780~GeV collected with the
BESIII detector, we report the first observation of the electromagnetic Dalitz
transition $h_c\to e^+e^-\eta_c$ with a statistical significance of
$5.4\sigma$. We measure the ratio of the branching fractions
$\frac{\mathcal{B}(h_c\rightarrow e^+e^-\eta_c)}{\mathcal{B}(h_c\rightarrow
\gamma \eta_c)}$ separately for the $h_c$ samples produced via
$\psi(3686)\to\pi^0h_c$ and $e^+e^-\to\pi^+\pi^-h_c$. The average ratio is
determined to be $(0.59\pm0.10(\text{stat.})\pm0.04(\text{syst.}))\%$, where
the uncertainty includes both statistical and systematic components.
