We analyze the elastic instability of torsional flow of a viscoelastic fluid in a cone-and-plate device with gap angle alpha. Assuming that the flow is axisymmetric and inertialess, we derive a model eigenvalue problem for the perturbation stream function valid for alpha much less than 1. The problem reduces to a singularly perturbed fourth-order boundary value problem related to the biharmonic operator. This problem is solved numerically and the results are compared with those obtained for radially unbounded domains. Assuming that the meniscus perturbation is small enough to preclude capillary instabilities, the flow is found to be slightly more stable than is predicted by the results for radially unbounded regions. (C) 2003 Elsevier B.V. All rights reserved.