Simultaneous photoluminescence and photocurrent measurements on individual single-walled carbon nanotubes reveal spontaneous dissociation of excitons into free electron-hole pairs. The correlation of luminescence intensity and photocurrent shows that a significant fraction of excitons are dissociating before recombination. Furthermore, the combination of optical and electrical signals also allows for extraction of the absorption cross section and the oscillator strength. Our observations explain the reasons why photoconductivity measurements in single-walled carbon nanotubes are straightforward despite the large exciton binding energies.