We consider steady flows of viscoelastic fluids for which the extrastress tensor is given by a differential constitutive equation and is such that the retardation time is large (weakly viscoelastic fluids).
We show the existence of a unique viscoelastic steady flow close to a given Newtonian flow and investigate its linear stability.
As an example, we consider the Bénard problem for viscoelastic fluids and we prove that there exists a nontrivial linearly stable flow of a weakly viscoelastic fluid in a container heated from below.