  We report measurements of $B \to \chi_{c1} \gamma K$ and $\chi_{c2} \gamma K$
decays using $772 \times 10^{6}$ $B\bar{B}$ events collected at the
$\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy
$e^+e^-$ collider. Evidence of a new resonance in the $\chi_{c1} \gamma$ final
state is found with a statistical significance of $3.8 \sigma$. This state has
a mass of $3823.1 \pm 1.8 {(stat)} \pm 0.7 {(syst)}$ MeV/$c^2$, a value that is
consistent with theoretical expectations for the previously unseen $1^3 D_2$
$c\bar{c}$ meson. We find no other narrow resonance and set upper limits on the
branching fractions of the $X(3872) \to \chi_{c1} \gamma$ and $\chi_{c2}
\gamma$ decays.
