We show how the optical properties of a single semiconductor quantum dot can be controlled with a small dc voltage applied to a gate electrode. We find that the transmission spectrum of the neutral exciton exhibits two narrow lines with approximately 2 mueV linewidth. The splitting into two linearly polarized components arises through an exchange interaction within the exciton. The exchange interaction can be turned off by choosing a gate voltage where the dot is occupied with an additional electron. Saturation spectroscopy demonstrates that the neutral exciton behaves as a two-level system. Our experiments show that the remaining problem for manipulating excitonic quantum states in this system is spectral fluctuation on a mueV energy scale.