  Recently, a new mechanism to generate a naturally small electroweak scale has
been proposed. It exploits the coupling of the Higgs to an axion-like field and
a long era in the early universe where the axion unchains a dynamical screening
of the Higgs mass. We present a new realization of this idea with the new
feature that it leaves no signs of new physics up to a rather large scale, 10^9
GeV, except for two very light and weakly coupled axion-like states. One of the
scalars can be a viable Dark Matter candidate. Such a cosmological Higgs-axion
interplay could be tested with a number of experimental strategies.
