  The recent discovery of the Higgs particle of 125 GeV has revised the
interest of the so-called Higgs factory, namely of a mu+mu- collider with
adequately cooled intensity of about 6 10^12 muons of each sign, a repetition
rate of 15-50 p/s and a luminosity of up to 10^32 cm-2 s-1. The process is the
direct production of a H_0 scalar in the s-state. Its partial widths can be
studied in clean conditions and with remarkable accuracy. The Ho mass itself
can be measured to about +-100 keV (deltaE/E <10^-6) in the (WW) channel with
the help of the decay frequency of the polarized mu->e nu nu decay electrons.
  The realization of a cascade of unconventional but very small rings of few
meters radius is described, in order to achieve the required longitudinal and
transverse emittances. Physics requirements and the many studies already
undertaken suggest that the next step, prior to but adequate for a H_0 physics
programme, could be the practical realization of a full scale muon cooling
demonstrator. The correct operation of the demonstrator may be initially
explored with the help of very low intensity muon beam already available in a
number of different accelerators.
  The additional but relatively conventional components necessary to realize
the facility with the appropriate luminosity should be constructed only after
the success of this initial cooling experiment has been conclusively
demonstrated. The ultimate mu+mu- collider for a Higgs Factory can be situated
within the existing CERN or FNAL sites.
