  If the $X(3872)$ is a weakly bound charm-meson molecule, it can be produced
in $e^+ e^-$ annihilation by the creation of $D^{*0} \bar D^{*0}$ from a
virtual photon followed by the rescattering of the charm-meson pair into $X$
and a photon. A triangle singularity produces a narrow peak in the cross
section for $e^+ e^- \to X \gamma$ about 2.2 MeV above the $D^{*0}
\bar{D}^{*0}$ threshold. We predict the normalized cross section in the region
near the peak. The peak from the triangle singularity may be observable by the
BESIII detector.
