We report the first observation of diffractive J/ψ(→μ+μ−) production in pp collisions at s√=1.8TeV. Diffractive events are identified by their rapidity gap signature. In a sample of events with two muons of transverse momentum pμT>2GeV/c within the pseudorapidity region |η|<1.0, the ratio of diffractive to total J/ψ production rates is found to be RJ/ψ=[1.45±0.25]%. The ratio RJ/ψ(x) is presented as a function of x-Bjorken. By combining it with our previously measured corresponding ratio Rjj(x) for diffractive dijet production, we extract a value of 0.59±0.15 for the gluon fraction of the diffractive structure function of the proton.