Reconstruction of B Meson Decays and Measurement of the B Quark and B Meson Production Cross-Sections at the Fermilab Tevatron Collider.

We report on the full reconstruction of B meson decays using data obtained at the Collider Detector at Fermilab in {| p}p collisions at sqrt s = 1.8 TeV. We have reconstructed B meson decays in the mode B_sp{u} {-}to J/psi+K^- (J/psi tomu^+mu^-). This sample represents the first and only full reconstruction of B mesons in a hadron collider. We use this sample to extract the B meson and b quark production cross sections. We obtain sigma=6.1+/-1.9(stat.)+/-2.4(syst.) mu b, for b quarks with transverse momentum P_{t}>10.5 GeV/c and rapidity | y|<1.0, and sigma=2.8+/-0.9(stat.)+/-1.1(syst.)mu b, for B_sp{u}{- } mesons with transverse momentum P _{t}>9.0 GeV/c and rapidity | y|<1.0. We compare the b quark result with an O(alpha^3) QCD calculation.