The stability properties of the laminar Kolmogorov flow of a viscoelastic Oldroyd-B fluid are investigated both analytically and numerically. Linear stability with respect to large-scale perturbations is studied by means of multiple-scale analysis. This technique yields an effective diffusion equation for the large-scale perturbation where the effective (eddy) viscosity can be computed analytically. Stability amounts to the positive definiteness of the eddy-viscosity tensor as a function of the Reynolds and the Deborah numbers. Two main results emerge from our analysis: (i) at small fluid elasticity, the flow is more stable than in the Newtonian case; (ii) at high elasticity, the flow is prone to elastic instabilities, occurring even at vanishing Reynolds number. The hypothesis of scale separation is verified up to moderate elasticity, as checked by numerical integration of the exact linearized equations by the Arnoldi method. Finally, it is shown that the addition of a stress diffusivity counteracts the effect of elasticity, in agreement with simple physical arguments.