The viscoelastic properties of thin polymer films confined between a sphere and a plane are investigated by use of a surface force apparatus designed as a rheometer operating at the molecular scale. Axial oscillatory motions yield a shear field from which a complex elastic modulus of the confined fluid can be deduced. As the sphere-plane distance is decreased, the fluid made of two adsorbed polymer layers and the free solution exhibits a liquid-solidlike transition at distance h(c). The measured complex modulus can be split into an elastic modulus G(c) accounting for the compression of the adsorbed chains and a complex shear modulus G(H)*(omega) expressing the draining process at the interface. The elastic modulus G(c) is directly connected to the gradient of static repulsive forces, and a compression modulus E(N) Of the adsorbed overlapping layers can be deduced from G(c). The flow term G(H)*(omega) is discussed in terms of viscoelastic parameters (zero-shear viscosity eta(0), average relaxation time tau(0)) as a function of sphere-plane distance h. For moderately compressed layers, a correlation length xi can be defined either from the compression modulus E(N) or from the zero-shear viscosity eta 0. Both lengths scale as (h(c) - h)(-49).