The transport properties of electronic devices are usually characterized on the basis of conductance measurements. Such measurements are adequate for devices in which transport occurs incoherently, but for very small devices-such as quantum dots(1,2)-the wave nature of the electrons plays an important role(3), Because the phase of an electron’s wavefunction changes as it passes through such a device, phase measurements are required to characterize the transport properties fully, Here we report the results of a double-slit interference experiment which permits the measurement of the phase-shift of an electron traversing a quantum dot, This is accomplished by inserting the quantum dot into one arm of an interferometer, thereby introducing a measurable phase shift between the arms, We find that the phase evolution within a resonance of the quantum dot can be accounted for qualitatively by a model that ignores the interactions between the electrons within the dot. Although these electrons must interact strongly, such interactions apparently have no observable effect on the phase. On the other hand, we also find that the phase behaviour is identical for all resonances, and that there is a sharp jump of the phase between successive resonance peaks. Adequate explanation of these features may require a model that includes interactions between electrons.