Two electrons on a string form a simple model system where Coulomb interactions are expected to play an interesting role. In the presence of strong interactions, these electrons are predicted to form a Wigner molecule, separating to the ends of the string. This spatial structure is believed to be clearly imprinted on the energy spectrum, yet so far a direct measurement of such a spectrum in a controllable one-dimensional setting is still missing. Here we use an ultraclean carbon nanotube to realize this system in a tunable potential. Using tunnelling spectroscopy we measure the addition spectra of two interacting carriers, electrons or holes, and identify seven low-energy states characterized by their exchange symmetries. The formation of a Wigner molecule is evident from a tenfold quenching of the fundamental excitation energy as compared with the non-interacting value. Our ability to tune the two-carrier state in space and to study it for both electrons and holes provides an unambiguous demonstration of this strongly interacting quantum ground state. © 2013 Macmillan Publishers Limited.