  The sensitivity to a light Higgs boson in the general 2HDM (II), with a mass
below 40 GeV, is estimated for an future e+e- linear collider operating with
very high luminosity at the Z peak (GigaZ). We consider a possible Higgs boson
production via the Bjorken process, the (hA) pair production, the Yukawa
process Z -> b {\bar b} h(A), -> tau {\bar tau} h(A), and the decay Z
->h(A)+gamma. Although the discovery potential is considerably extended
compared to the current sensitivities, mainly from LEP, the existence of a h or
A even with a mass of a few GeV cannot be excluded with two billion Z decays.
The need to study the very light Higgs scenario at a linear e+e- collider
running at several hundred GeV and the LHC is emphasised.
