This Letter presents the observation of the rare Z boson decay Z -> psi l(+)l(-). Here, psi represents contributions from direct J/psi and psi(2S) -> J/psi X, l(+)l(-) 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) -> J/psi X contribution, the ratio of the branching fraction of the exclusive decay Z -> J/psi X, l(+) l(-) to the decay Z -> mu(+)mu(-)mu(+)mu(-) within a fiducial phase space is measured to be B(Z -> J/psi X, l(+) l(-))/B(Z -> mu(+)mu(-)mu(+)mu(-)) = 0.67 +/- 0.18(stat) +/- 0.05(syst).
