We describe a highly improved atomic force microscope for quantitative nanomechanical measurements in liquids. The main feature of this microscope is a modified fiber interferometer mounted on a five axis inertial slider which provides a deflection sensitivity that is significantly better than conventional laser deflection based systems. The measured low noise floor of 572.0 fmHz provides excellent cantilever amplitude resolution. This allows us to operate the instrument far below resonance at extremely small cantilever amplitudes of less than 1 Å. Thus linear measurements of nanomechanical properties of liquid systems can be performed. In particular, we present measurements of solvation forces in confined octamethylcyclotetrasiloxane and water with amplitudes smaller than the size of the respective molecules. In general, the development of the instrument is important in the context of quantitative nanomechanical measurements in liquid environments.