The analytical solution for the steady-state flow in a pipe of viscoelastic fluids obeying the complete Phan-Thien-Tanner constitutive equation with a linear stress coefficient is derived. The results include the radial profiles of all relevant stresses, of the axial velocity and of the viscosity. The pipe flow is found to be unstable when the pressure gradient exceeds a critical value determined by a maximum shear rate at the wall. Expressions are also given for the viscometric viscosity and the first and second normal stress difference coefficients, as a function of the shear rate, in steady plane shear flow. For this case, and in line with similar results for the Johnson-Segalman model, the shear stress was found not to be a monotonically increasing function of the shear rate as strong shear-thinning sets in. A new finding is that the critical condition for the maximum shear stress in the simple-shear case is related to the condition for existence of steady-state solution in the pipe flow case. While this may seem evident “a priori” since the pipe flow is a viscometric flow, it is formally demonstrated.