We report on the observation of long-range forces between ultracold rubidium atoms that are mutually coupled by the field of a driven high-finesse optical cavity. Even for much less than one photon in the cavity on average, the forces strongly influence the spatial distribution of the atoms. This manifests itself as an asymmetric normal-mode spectrum of the strongly coupled atoms-cavity system. Expressions are given for the dipole force and the diffusion coefficient for the atoms in the presence of the other atoms. The data agree well with calculated spectra, which include the full motional dynamics of the many- atom system.