By analyzing a data set of 2.92 fb$^{-1}$ of $e^+e^-$ collision data taken at $\sqrt s= 3.773~\rm GeV$ and 106.41$\times 10^{6}$ $\psi(3686)$ decays taken at $\sqrt s= 3.686~\rm GeV$ with the BESIII detector at the BEPCII collider, we measure the branching fraction and the partial decay width for $\psi(3770)\to\gamma\chi_{c0}$ to be ${\mathcal B}(\psi(3770)\to\gamma\chi_{c0})=(6.88\pm0.28\pm0.67)\times 10^{-3}$ and $\Gamma[\psi(3770)\to\gamma\chi_{c0}]=(187\pm8\pm19)~\rm keV$, respectively. These are the most precise measurements to date.