The eta_b (1S) is the JPC = 0-+ bb-bar ground state, and has
not yet been experimentally observed.  Theorists have recently suggested
that it could be observed at the Fermilab Tevatron through its decay to
J/psi J/psi, if not in the 1992-96 ("Run 1") dataset, then in Run 2.  This
article describes a search for this particle at CDF is Run 1 using this
decay channel.  A small cluster is seen, with 7 events where 1.8 events are
expected from background.  The statistical significance is estimated to be
2.2 sigma, and an upper limit is set on the product of cross section and
branching fractions.