The purely elastic instability of torsional flow a viscoelastic fluid confined between two coaxial parallel plates is considered. This problem is analyzed in the limit of small aspect ratio α = h/a, and zero Reynolds number where h is the plate separation and a is the plate radius. In this limit we derive a set of equations which can be solved analytically for the Oldroyd-B model. Our analysis shows that stability is controlled by a local elasticity number ε≡rλ2ϖ2/α, where r is the local radius, γ is the relaxation time and ϖ is the constant rotation rate. For ε less than some critical value εc the flow is linearly stable. As ε increases past this critical value the flow loses stability to a time dependent oscillatory mode. The most dangerous mode has a short wave length of the order O(α). Our results are in good agreement with those of Oztekin and Brown (1993) and Byars et al. (1994). A criterion to ensure stability to infinitesimal disturbances is proposed.