  We report a measurement of the branching fraction of
$\psi(3770)\to\gamma\chi_{c1}$ and search for the transition
$\psi(3770)\rightarrow \gamma \chi_{c2}$ based on 2.92~fb$^{-1}$ of $e^+e^-$
data accumulated at $\sqrt{s}=3.773$~GeV with the BESIII detector at the BEPCII
collider. The measured branching fraction of $\psi(3770) \rightarrow \gamma
\chi_{c1}$ is $\mathcal B(\psi(3770) \rightarrow \gamma \chi_{c1})=(2.48 \pm
0.15 \pm 0.23) \times 10^{-3}$, which is the most precise measurement to date.
The upper limit on the branching fraction of $\psi(3770)\rightarrow \gamma
\chi_{c2}$ at a $90\%$ confidence level is $\mathcal B(\psi(3770) \rightarrow
\gamma \chi_{c2})<0.64 \times 10^{-3}$. The corresponding partial widths are
$\Gamma(\psi(3770) \to \gamma \chi_{c1}) =(67.5\pm 4.1\pm 6.7)$~keV and
$\Gamma(\psi(3770) \to \gamma \chi_{c2}) < 17.4$~keV.
