A search for Z boson decays to pairs of b-quark jets has been
performed in the full dataset collected with the CDF detector at the Tevatron
ppbar collider.  After the selection of a pure sample of bb-bar events by
means of the identification of secondary vertices from b-quark decays, we have
used two kinematic variables to further discriminate the electroweak bb-bar
production from QCD processes, and sought evidence for the Z decay in the
dijet invariant mass distribution.  An absolute background prediction allows
the extraction of an excess of events inconsistent with the background
predictions by 3.23 sigma but in good agreement with the amount and
characteristics of the expected signal.  We then fit the mass distribution
with an unbinned likelihood technique, and obtain a Z -> bb-bar signal
amounting to 91 +/- 30 +/- 19 events.