Summary: The linear stability of the inertialess, pressure-driven Poiseuille flow on an Oldroyd-B fluid through a slightly curved channel is considered. The flow is shown to be unstable in certain flow parameter regimes. The critical conditions and the structure of the vortex flow at the onset of instability are presented. These results reveal that there is a purely elastic, instability in the flow, and the instability is a stationary mode in contrast to the elastic, oscillatory instability that occurs in Taylor-Couette flow. In addition, the mechanism of the instability is investigated through an examination of the disturbance-energy equation.