The letter reports the full reconstruction of the B messons through decay chain
B+- --> J/psiK+-, J/psi --> mu+ mu-, using data obtained at the Collider
Detector at Fermilab in p anti-p collisions s**(1/2) = 1.8 TeV.  This
exclusive sample, the first observed at a hadron collider, is then used to
measure the B-meson cross section, from which we extract the b-quark cross
section.  We obtain sigma = 2.8+-0.9(stat)+-1.1(syst) microb for B- mesons
with Pt > 9.0 GeV/c and rapidity |y| < 1.0. We obtain sigma = 6.1+-1.9(stat)
+-2.4(syst) microb, for b quarkswith transverse momentum Pt > 11.5 Gev/c and
rapidity |y| < 1.0.