  We present spectroscopic observations obtained with the FORS1 instrument on
the ESO VLT under good seeing conditions of the radio-quiet z_em = 3.04 quasar
Q1205-30 and its associated extended Lyman-alpha emission. The extended
Lyman-alpha emission was originally found in a deep narrow band image targeting
a z_abs = z_em Lyman-limit system in the spectrum of the QSO. Using spectral
point-spread function fitting to subtract the QSO spectrum, we clearly detect
the extended Lyman-alpha emission as well as two foreground galaxies at small
impact parameters (2.12 +- 0.04 and 2.77 +- 0.07 arcsec). The redshifts of the
two foreground galaxies are found to be z = 0.4732 and z = 0.865. We determine
the redshift and velocity profile for the extended Lyman-alpha emission, and
analyzing the velocity offsets between eight QSO emission lines we refine the
quasar redshift determination. We use the new redshifts to infer the geometry
of the complex. We find that the extended Lyman-alpha emission is clearly
associated with the quasar. A Lyman-alpha luminosity of 5.6 x 10^43 erg s^-1
places this extended emission at the high luminosity end of the few previous
detections around radio-quiet quasars. The extended Lyman-alpha emission is
best explained by hydrogen falling into the dark matter halo inhabited by the
quasar.
