The stability of an infinite viscoelastic plate on an elastic foundation in a viscous incompressible flow is studied. The Navier-Stokes system is linearized for an exponential velocity profile. The problem is reduced by a Fourier-Laplace transform to a system of ordinary differential equations, whose solution is found in the form of convergent series. The roots of the dispersion relation that characterize the stability of the system are found numerically. The effect of the viscosities of the fluid and the plate on the stability of the waves propagating upstream and downstream is studied. The results are compared with available data on the stability of a viscoelastic plate in an ideal fluid flow.