  We report the observation of the $X(3823)$ in the process $e^+e^-\to
\pi^+\pi^-X(3823) \to \pi^+\pi^-\gamma\chi_{c1}$ with a statistical
significance of $6.2\sigma$, in data samples at center-of-mass energies
$\sqrt{s}=$4.230, 4.260, 4.360, 4.420 and 4.600~GeV collected with the BESIII
detector at the BEPCII electron positron collider. The measured mass of the
$X(3823)$ is $(3821.7\pm 1.3\pm 0.7)$~MeV/$c^2$, where the first error is
statistical and the second systematic, and the width is less than $16$~MeV at
the 90\% confidence level. The products of the Born cross sections for
$e^+e^-\to \pi^+\pi^-X(3823)$ and the branching ratio $\mathcal{B}[X(3823)\to
\gamma\chi_{c1,c2}]$ are also measured. These measurements are in good
agreement with the assignment of the $X(3823)$ as the $\psi(1^3D_2)$ charmonium
state.
