The Linear-response effective shear moduli of polymer melts confined between strongly adsorbing surfaces (parallel plates of mica) was studied as a function of the excitation frequency. Linear response (achieved with shear amplitudes of approximate to 2 Angstrom) implies that measurements did not perturb the film structure. The measurements employed a surface forces apparatus modified for dynamic mechanical shear oscillation. The polymers (atactic poly(phenylmethylsiloxane), PPMS, chain length from 31 to 153 skeletal bonds) were selected to be glass-forming with a low glass transition temperature, in order to avoid issues of possible surface-induced crystallization which have been much discussed. Three principal conclusions emerged. First, in a comparison of polymers of different molecular weight, both the shear moduli (measured at fixed film thickness) and the static normal forces (to compress the polymers to this film thickness) scaled approximately with the estimated unperturbed radius of gyration of the chain (R(G)). Second, a transition from fluid- to solidlike response occurred when the film thickness was less than 5-6 R(G) : the longest relaxation time slowed to the point that the storage shear modulus (G’) exceeded the loss shear modulus (G") over the entire frequency spectrum. This contrasts strongly with the behavior of the bulk samples, for which Rouse dynamics would be expected. Third, the magnitude of G’ under conditions of strong confinement was characteristic of rubberlike elasticity, approximately 10(5) N m(-2) for the range of chain length that was studied. This indicates enhanced entanglement interactions in thin polymer films. Two accompanying papers analyze the nonlinear theology and the respective influences of surface adsorption and of geometrical confinement.