Global linear stability analysis of the creeping flow of an Oldroyd-B liquid, confined between eccentric cylinders co-rotated at equal angular speeds Omega, is performed using a submatrix-based transformation algorithm [K. Arora, R. Sureshkumar, J. Non-Newtonian Fluid Mech. 104 (2002) 75]. The eccentricity parameters is defined as the ratio of the distance between the cylinder centers to the average gap width d. In the limit as epsilon -> 0 and for narrow gaps, the base flow corresponds to a solid body rotation. A flow instability is predicted even when epsilon << 1. For sufficiently large values of the solvent to total viscosity ratio beta, the most dangerous disturbance is time-periodic with frequency approximate to 0.1 Omega/delta, where delta denotes the ratio of d to the inner cylinder radius. The critical Weissenberg number We(c) defined as the product of the fluid relaxation time and the characteristic shear rate at the onset, obeys a scaling law of the form We(c)epsilon(2)delta(1/2) = K where K is an O(1) constant that is dependent on beta. This scaling is explained based on the effect of the variation in epsilon on the convection of the stress perturbations by the base flow. Predictions of We(c) are in qualitative agreement with the onset Weissenberg number for a time-dependent secondary flow experimentally reported for non-shear thinning viscoelastic polymer solutions [I.M. Dris, E.S.G. Shaqfeh, J. Non-Newtonian Fluid Mech. 80 (1998) 1]. (c) 2005 Elsevier BX All rights reserved.