The first flow regimes which have been observed experimentally for a circular Couette flow with a stable, axial stratification in density are investigated through direct numerical simulations of the three-dimensional Navier-Stokes equations for a Boussinesq fluid. The setup of two concentric cylinders has a nondimensional gap width of epsilon=(b-a)/a=0.289; the outer cylinder is tired and the stratification in density in the axial direction is linear. The main effect of an axial density stratification is to reduce the height of the Taylor vortices and to cause the formation of density layers of small aspect ratio. For large enough Prandtl number, the primary bifurcation from circular Couette flow is found to be axisymmetric and of Hopf-type in the direct numerical simulations. An analytical solution for onset of instability and slightly different boundary conditions from the experimental ones agrees within 0.6% with numerical simulations at a Prandtl number of 700. The experimental flow regimes with well-defined density layers are well reproduced by the numerical simulations in the appropriate range of relative Reynolds number Re/Re-c1, where Re-c1 denotes the critical Reynolds number for the primary bifurcation from circular Couette flow. However, the increase of axial scale with Re/Re-c1 is found to be continuous, whereas it is quantized in the laboratory experiments. Numerical results reveal that the first two transitions between the flow regimes are primarily due to the temporal behavior of the axially symmetric part of the flow. Onset of nonaxisymmetric motions appears at the same Re/Re(c1)approximate to 1.18 as in the homogeneous fluid case at the same eta=a/b. Stratification precludes large axial displacements and the azimuthal modes patterns have a quite distinct appearance from the homogeneous wavy modes. At large enough Re, a destabilization of the jet-like outflow between pairs of vortices causes the suppression of the density front which is located at the same axial height. This nonaxisymmetric flow regime presents common features with the wavy outflow boundary (WOE) pattern, which is commonly observed in the homogeneous Couette-Taylor case.