The observation of the bottom, strange baryon Ξ(b)(0) through the decay chain Ξ(b)(0)→Ξ(c)(+)π-, where ΞΞ(c)(+)→Ξ- π+ π+, Ξ-→Λπ-, and Λ→pπ-, is reported by using data corresponding to an integrated luminosity of 4.2 fb(-1) from pp collisions at square root(s)=1.96 TeV recorded with the Collider Detector at Fermilab. A signal of 25.3(-5.4)(+5.6) candidates is observed whose probability of arising from a background fluctuation is 3.6×10(-12), corresponding to 6.8 gaussian standard deviations. The Ξ(b)(0) mass is measured to be 5787.8±5.0(stat)±1.3(syst) MeV/c2. In addition, the Ξ(b)- baryon is observed through the process Ξ(b)-→Ξ(c)(0)π-, where Ξ(c)(0)→Ξ- π+, Ξ-→Λπ-, and Λ→pπ-.