The steady flow of an incompressible Oldroyd 8-constant fluid in the annular region between two concentric cylinders, or so-called cylindrical Couette flow, is investigated. The inner cylinder rotates with an angular velocity W about its own axis, z-axis, while the outer cylinder is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum equation. An analytical solution is obtained through the expansion of the dynamical variables in power series of the dimensionless retardation time. The primary velocity term denotes the Newtonian rotation about the z-axis. The first-order is a vanishing term. The second-order results in a secondary flow represented by the stream-function. This second-order term is the viscoelastic contribution to the primary viscous flow. The second-order approximation depends on the four viscometric parameters of the fluid.