The stability of plane Poiseuille flow to infinitesimal perturbations was studied for the second order and Maxwell fluid rheological models. When the Deborah number based disturbance propagation is small the second order fluid is a consistent constitutive equation and the two models give identical results. This occurs for elasticity number (E) less than 5×10 −4 . At higher values of E the second order fluid cannot be used. The critical Reynolds number is a slowly decreasing function of E up to E ≈10 −4 and a rapidly decreasing function subsequently in the region where fluid relaxation effects become important. For a sufficiently elastic fluid the flow transition is governed by a new mode of the Orr‐Sommerfeld equation and differs qualitatively from that for a Newtonian liquid. The results suggest the likelihood of low Reynolds numberinstability in highly elastic liquids.