The flow in a pipe of wormlike micellar solutions is examined using a simple model that consists of the codeformational Maxwell constitutive equation and a kinetic equation that accounts for the breaking and reformation of micelles. The model needs six parameters, all, of which are extracted from single independent theological experiments. One of the parameters, the shear-banding intensity parameter is associated with the stress plateau in the shear-banding region. The stress plateau is set in our model by the criterion of equal extended Gibbs free energy of the bands. The model predicts a Newtonian (parabolic profile) flow at low-shear rates or low-pressure gradients, followed by shear thinning up to a critical rate where instabilities and long transients appear. At this critical shear rate, a shear-banding flow region arises near the pipe wall. The model indicates that tube lengths up to 400 diameters are required to obtain fully developed flow, where a pluglike profile at the center of the tube coexists with a region supporting a much higher shear rate next to the wall. Shear-banding flow is present up to a second critical shear rate. At shear rates larger than the second critical rate, the parabolic velocity profile is recovered, except near the center of the tube where a small shear-banding flow region remains because the stress at that radial position is equal to the plateau stress. This is a consequence of the linear dependence of the shear stress with the pipe radius. The predictions of the model are compared with experimental results from the literature.