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).