Resonant Rayleigh scattering from self-assembled one-dimensional molecular J-aggregate wires reveals a distinct dependence of the exciton energy on the width of lateral extension. For the J aggregates used in this study, strong in-line dipole coupling leads to a delocalization of the exciton wave function over several molecular units. Polarization dependent measurements of resonantly scattered light from the wires show that the exciton dipole moment is oriented perpendicular to the long axis. The experimental observations can be described by applying a quantization condition to the center of mass motion of the J-band exciton in the wires.