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^2m_{A}