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Stress and Stability analysis of Viscoelastic rotating flow is studied in the narrow gap limit. The Galerkin projection method is used to derive dynamical System. Flow parameters are obtained using Mathematica Software. Stresses are computed in a wide range of the fluid elasticity effects. It is found that, all stress components decreased as fluid elasticity increased. Also the influence of fluid elasticity on the stability Taylor vortices is examined for the flow. It is observed that, the fluid elasticity tends to precipitate the onset of Taylor vortices. However, unlike the fluid elasticity, fluid retardation tends to delay the onset of Taylor vortices.