In this paper, a new analytical solution for viscoelastic flow in curved elliptical pipes is presented for the first time. The perturbation method is used to derive the analytical solution, and the curvature ratio is considered as the perturbation parameter. The Oldroyd-B model is used as the constitutive equation, so the result of the present study could be useful for modeling the flow of dilute polymeric solutions inside curved elliptical pipes. The analytical solution is derived using an appropriate transformation that converts the elliptical shape of a cross section to the unit circle. The transformed governing equations are solved to the second-order terms using the perturbation method and the velocity field is obtained by implementing the inverse transformation on the results. Here, the effects of geometry, Weissenberg number and Reynolds number on the axial velocity, secondary flows and flow rate are studied in detail. (C) 2013 Elsevier B.V. All rights reserved.