An exact analytical solution for a Poiseuille flow of a Giesekus fluid in an annulus is obtained, taking into account the effects of the non-linearity of the constitutive equation. The fluid velocity is given in terms of Legendre elliptic integrals and Jacobi elliptic functions. The fluid behaviour is described using the Deborah number along the mobility factor and the radius ratio, while the tangential and the normal stresses are also evaluated. The shear-rate, the velocity and the stress tensor components as function of the involved parameters are graphically represented and analysed in detail.