  At redshifts z >~ 2, most of the baryons reside in the smooth intergalactic
medium which is responsible for the low column density Lyman-alpha forest. This
photoheated gas follows a tight temperature-density relation which introduces a
cut-off in the distribution of widths of the Lyman-alpha absorption lines
(b-parameters) as a function of column density. We have measured this cut-off
in a sample of nine high resolution, high signal-to-noise quasar spectra, and
determined the thermal evolution of the intergalactic medium in the redshift
range 2.0-4.5. At redshift z ~ 3, the temperature at the mean density shows a
peak and the gas becomes nearly isothermal. We interpret this as evidence for
the reionization of HeII.
