  We obtained optical spectra of the counterpart of the ultraluminous X-ray
source NGC 5408 X-1 using the FORS spectrograph on the VLT. The spectra show
strong high excitation emission lines, He{\sc ii} $\lambda$4686 and [Ne{\sc V}]
$\lambda$3426, indicative of X-ray photoionization. Using the measured X-ray
spectrum as input to a photoionization model, we calculated the relation
between the He{\sc ii} and X-ray luminosities and found that the He{\sc ii}
flux implies a lower bound on the X-ray luminosity of $3 \times 10^{39} \rm erg
s^{-1}$. The [Ne{\sc v}] flux requires a similar X-ray luminosity. After
subtraction of the nebular emission, the continuum appears to have a power-law
form with a spectral slope of $-2.0^{+0.1}_{-0.2}$. This is similar to low-mass
X-ray binaries where the optical spectra are dominated by reprocessing of
X-rays in the outer accretion disk. In one observation, the continuum, He{\sc
ii} $\lambda$4686, and [Ne{\sc V}] $\lambda$3426 fluxes are about 30% lower
than in the other five observations. This implies that part of the line
emission originates within one light-day of the compact object. Fitting the
optical continuum emission and archival X-ray data to an irradiated disk model,
we find that $(6.5 \pm 0.7) \times 10^{-3}$ of the total bolometric luminosity
is thermalized in the outer accretion disk. This is consistent with values
found for stellar-mass X-ray binaries and larger than expected in models of
super-Eddington accretion flows. We find no evidence for absorption lines that
would permit measurement of the radial velocity of the companion star.
