  We report on the investigation of the rare decay
$\eta\rightarrow\pi^0+\text{e}^++\text{e}^-$ which is of interest to study both
$C$ violation in the electromagnetic interaction and to search for
contributions from physics beyond the Standard Model, since the allowed decay
via a two-photon intermediate state is strongly suppressed. The experiment has
been performed using the WASA-at-COSY installation, located at the COSY
accelerator of the Forschungszentrum J\"ulich, Germany. In total $3\times10^7$
events of the reaction $\text{p}+\text{d}\rightarrow{^3\text{He}}+\eta$ have
been recorded at an excess energy of $Q = 59.8\,\textrm{MeV}$. Based on this
data set the $C$ parity violating decay
$\eta\rightarrow\pi^0+\gamma^*\rightarrow\pi^0+\textrm{e}^++\textrm{e}^-$ via a
single-photon intermediate state has been searched for, resulting in new upper
limits of $\Gamma\left(\eta\rightarrow\pi^0+\textrm{e}^++\textrm{e}^-\right)/
\Gamma\left(\eta\rightarrow\pi^++\pi^-+\pi^0\right) < 3.28\times10^{-5}$ and
$\Gamma\left(\eta\rightarrow\pi^0+\textrm{e}^++\textrm{e}^-\right)/
\Gamma\left(\eta\rightarrow\textrm{all}\right) < 7.5\times10^{-6}\ (\textrm{CL}
= 90\,\%)$, respectively.
