The present work proposes an exact analytical solution for a Couette flow of a Giesekus fluid in an annulus, taking into account the effects of the non-linearity of the constitutive equation. The fluid velocity is given as a function of the Deborah number, the mobility factor and the radius ratio. Analysis shows that if the mobility factor is greater than 0.5, there exists a limit (maximum) Deborah number above which the studied phenomenon has no valid solution. The normal, the tangential and the azimuthal stresses are evaluated; the ratio between the torque required to rotate the inner cylinder for a Giesekus and for a Newtonian fluid is investigated. Results show that an increase of the fluid elasticity and/or of the mobility factor, leads to a decrease of the friction factor ratio for all values of the radius ratio. Furthermore, the behaviour of the velocity and of the stress tensor components as function of the involved parameters is graphically represented and discussed in detail. (C) 2015 Elsevier B.V. All rights reserved.