The classical energy method of Orr is used to investigate the stability of a plane Couette flow of a viscoelastic fluid obeying the constitutive equation of the so‐called second‐order fluid. For two‐dimensional disturbances of any magnitude, the presence of elasticity has been found to stabilize the flow. For small three‐dimensional disturbances invariant in the flow direction, the presence of elasticity has been found to affect the stability either way, depending on the relative values of the constants in the constitutive equation.