The stability of the flow of an incompressible, viscous fluid through a pipe of circular cross-section, curved about a central axis is investigated in a weakly nonlinear regime. A sinusoidal pressure gradient with zero mean is imposed, acting along the pipe. A WKBJ perturbation solution is constructed, taking into account the need for an inner solution in the vicinity of the outer bend, which is obtained by identifying the saddle point of the Taylor number in the complex plane of the cross-sectional angle coordinate. The equation governing the nonlinear evolution of the leading-order vortex amplitude is thus determined. The stability analysis of this flow to axially periodic disturbances leads to a partial differential system dependent on three variables, and since the differential operators in this system are periodic in time, Floquet theory may be applied to reduce it to a coupled infinite system of ordinary differential equations, together with homogeneous uncoupled boundary conditions. The eigenvalues of this system are calculated numerically to predict a critical Taylor number consistent with the analysis of Papageorgiou (1987). A discussion of how nonlinear effects alter the linear stability analysis is also given. It is found that solutions to the leading-order vortex amplitude equation bifurcate subcritically from the eigenvalues of the linear problem.