Electron holography performed in situ inside a high resolution transmission electron microscope has been used to determine the magnitude and spatial distribution of the electric field surrounding individual field-emitting carbon nanotubes. The electric field (and hence the associated field emission current) is concentrated precisely at the tips of the nanotubes and not at other nanotube defects such as sidewall imperfections. The electric field magnitude and distribution are stable in time, even in cases where the nanotube field emission current exhibits extensive temporal fluctuations.