  A sample of $1.69\times 10^7$ fully reconstructed $\pi^0\to\gamma e^+e^-$
decay candidates collected by the NA48/2 experiment at CERN in 2003--2004 is
analysed to search for the dark photon ($A'$) production in the $\pi^0\to\gamma
A'$ decay followed by the prompt $A'\to e^+e^-$ decay. No signal is observed,
and an exclusion region in the plane of the dark photon mass $m_{A'}$ and
mixing parameter $\varepsilon^2$ is established. The obtained upper limits on
$\varepsilon^2$ are more stringent than the previous limits in the mass range
$9~{\rm MeV}/c^2<m_{A'}<70~{\rm MeV}/c^2$. The NA48/2 sensitivity to the dark
photon production in the $K^\pm\to\pi^\pm A'$ decay is also evaluated.
