We report the full reconstruction of χc mesons through the decay chain χc→J/ψ γ, J/ψ→μ+μ−, using data obtained at the Collider Detector at Fermilab in 2.6±0.2 pb−1 of p¯p collisions at √s =1.8 TeV. This exclusive χc sample is used to measure the χc-meson production cross section times branching fractions. We obtain σ×B=3.2±0.4(stat)−1.1+1.2(syst) nb for χc mesons decaying to J/ψ with pT>6.0 GeV/c and pseudorapidity ‖η‖<0.5. From this and the inclusive J/ψ cross section we calculate the inclusive b-quark cross section to be 12.0±4.5 μb for pTb>8.5 GeV/c and ‖yb‖<1.