  We report the first observation of diffractive $J/\psi(\to \mu^+\mu^-)$
production in $\bar pp$ collisions at $\sqrt{s}$=1.8 TeV. Diffractive events
are identified by their rapidity gap signature. In a sample of events with two
muons of transverse momentum $p_T^{\mu}>2$ GeV/$c$ within the pseudorapidity
region $|\eta|<$1.0, the ratio of diffractive to total $J/\psi$ production
rates is found to be $R_{J/\psi}= [1.45\pm 0.25]%$. 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 $0.59\pm 0.15$ for the gluon fraction of the diffractive
structure function of the proton.
