  With data samples collected with the BESIII detector operating at the BEPCII
storage ring at center-of-mass energies from 4.009 to 4.420 GeV, the process
$e^{+} e^{-} \to \gamma X(3872)$ is observed for the first time with a
statistical significance of $6.3\sigma$. The measured mass of the $X(3872)$ is
($3871.9\pm 0.7_{\rm stat.}\pm 0.2_{\rm sys.}$) MeV/$c^2$, in agreement with
previous measurements. Measurements of the product of the cross section
$\sigma[e^{+} e^{-} \to \gamma X(3872)]$ and the branching fraction
$\mathcal{B}[X(3872) \to \pi^{+} \pi^{-} J/\psi]$ at center-of-mass energies
4.009, 4.229, 4.260, and 4.360 GeV are reported. Our measurements are
consistent with expectations for the radiative transition process $Y(4260) \to
\gamma X(3872)$.
