Single-wall carbon nanotubes(1,2) are ideally suited for electron-transport experiments on single molecules because they have a very robust atomic and electronic structure and are sufficiently long to allow electrical connections to lithographically defined metallic electrodes. The electrical transport properties of single nanotubes(3) and bundles of nanotubes(4) have so far been interpreted by assuming that individual electrons within the nanotube do not interact, an approximation that is often well justified for artificial mesoscopic devices such as semiconductor quantum dots(5). Here we present transport spectroscopy data on an individual carbon nanotube that cannot be explained by using independent-particle models and simple shell-filling schemes. For example, electrons entering the nanotube in a low magnetic field are observed to all have the same spin direction, indicating spin polarization of the nanotube. Furthermore, even when the number of electrons on the nanotube is fixed, we find that variation of an applied gate voltage can significantly change the electronic spectrum of the nanotube and can induce spin flips. The experimental observations point to significant electron-electron correlations. We explain our results phenomenologically using a model that assumes that the capacitance of the nanotube depends on its many-body quantum state.