We report on the experimental observation of excitonic molecules (biexcitons) in high-quality V-shaped quantum wires. By means of spatially resolved near-field photoluminescence spectroscopy, first we unambiguously isolate excitons with a one-dimensional character, and secondly we present the optical transition from 1D biexciton to 1D exciton as a function of excitation power density. The density of these biexcitonic molecules is shown to increase superlinearly compared to the linear variation of exciton density.