We report the first observation of diffractive J/psi(--> mu+mu-) production in pp collisions at root square[s] = 1.8 TeV. Diffractive events are identified by their rapidity gap signature. In a sample of events with two muons of transverse momentum p(mu)T>2 GeV/c within the pseudorapidity region /eta/ < 1.0, the ratio of diffractive to total J/psi production rates is found to be R(J/psi) = [1.45 +/- 0.25]%. The ratio R(J/psi)(x) is presented as a function of x-Bjorken. By combining it with our previously measured corresponding ratio R(jj)(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.