This Letter presents the observation of the rare Z boson decay Z→ψℓ+ℓ−. Here, ψ represents contributions from direct J/ψ and ψ(2S)→J/ψX, ℓ+ℓ− is a pair of electrons or muons, and the J/ψ meson is detected via its decay to μ+μ−. 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 ψ(2S)→J/ψX contribution, the ratio of the branching fraction of the exclusive decay Z→J/ψℓ+ℓ− to the decay Z→μ+μ−μ+μ− within a fiducial phase space is measured to be B(Z→J/ψℓ+ℓ−)/B(Z→μ+μ−μ+μ−)=0.67±0.18(stat)±0.05(syst).