We report the observation of the bottom, doubly-strange baryon Omega(-)(b) through the decay chain Omega(-)(b)-> J/psi Omega(-), where J/psi ->mu(+)mu(-), Omega(-)->Lambda K(-), and Lambda -> p pi(-), using 4.2 fb(-1) of data from p (p) over bar collisions at root s = 1.96 TeV, and recorded with the Collider Detector at Fermilab. A signal is observed whose probability of arising from a background fluctuation is 4.0x10(-8), or 5.5 Gaussian standard deviations. The Omega(-)(b) mass is measured to be (6054.4 +/- 6.8(stat) +/- 0.9(syst)) MeV/c(2). The lifetime of the Omega(-)(b) baryon is measured to be (1.13(-0.40)(+0.53)(stat) +/- 0.02(syst)) ps. In addition, for the Xi(-)(b) baryon we measure a mass of (5790.9 +/- 2.6(stat) +/- 0.8(syst)) MeV/c(2) and a lifetime of (1.56(-0.25)(+0.27)(stat) +/- 0.02(syst)) ps. Under the assumption that the Xi(-)(b) and Omega(-)(b) are produced with similar kinematic distributions to the Lambda(0)(b) baryon, we find (sigma(Xi(-)(b))xB(Xi(-)(b)-> J/psi Xi(-)))/(sigma(Lambda(0)(b))xB(Lambda(0)(b)-> J/psi Lambda)) = 0.167(-0.025)(+0.037)(stat) +/- 0.012(syst) and (sigma(Omega(-)(b))xB(Omega(-)(b)-> J/psi Omega(-)))/(sigma(Lambda(0)(b))xB(Lambda(0)(b)-> J/psi Lambda)) = 0.045(-0.012)(+0.017)(stat) +/- 0.004(syst) for baryons produced with transverse momentum in the range of 6-20 GeV/c.