The viscoelastic flow of an Upper Convected Maxwell fluid, confined between two infinitely long eccentric rotating cylinders, is investigated. The two-dimensional steady-state flow as well as the stability of the flow against true three-dimensional (azimuthally and axially periodic and radially non-periodic) disturbances is analyzed numerically using pseudospectral methods. In this numerical scheme, variables are expressed as Fourier series in the periodic direction and as Chebyshev polynomials in the radial direction. The linear stability analysis shows that the critical Reynolds number, corresponding to the onset of flow instability, increases with eccentricity for a Newtonian flow. The critical wavenumber in the axial direction is found to remain nearly constant in the eccentricity range between 0 and 0.5. Addition of small flow elasticity in high Re flows is found to destabilize the system, causing the critical wavenumbers to slightly increase. For a purely elastic flow, it is found that the critical Deborah number decreases with an increase in eccentricity and the critical wavenumber also decreases for the parameters examined in our study.