This Letter presents the observation of the rare Z boson decay Z $\to\psi\ell^+\ell^-$. Here, $\psi$ represents contributions from direct J/$\psi$ and $\psi$(2S) $\to$ J/$\psi X$, $\ell^+\ell^-$ is a pair of electrons or muons, and the J/$\psi$ meson is detected via its decay to $\mu^+\mu^-$. The sample of proton-proton collision data, collected by the CMS experiment at the LHC at a center-of-mass energy of 13 TeV, corresponds to an integrated luminosity of 35.9 fb$^{-1}$. The signal is observed with a significance in excess of 5 standard deviations. After subtraction of the $\psi$(2S) $\to$ J/$\psi X$ contribution, the ratio of the branching fraction of the exclusive decay Z $\to\psi\ell^+\ell^-$ to the decay Z $\to\mu^+\mu^-\mu^+\mu^-$ within a fiducial phase space is measured to be $\mathcal{B}($Z $\to\psi\ell^+\ell^-) / \mathcal{B}($Z $\to\mu^+\mu^-\mu^+\mu^-) =$ 0.67 $\pm$ 0.18 (stat) $\pm$ 0.05 (syst).