  We present a class of simple scalar-tensor models of gravity with one scalar
field (dilaton $\Phi$) and only one unknown function (cosmological potential
$U(\Phi)$). These models might be considered as a stringy inspired ones with
broken SUSY. They have the following basic properties: 1) Positive dilaton
mass, $m_\Phi$, and positive cosmological constant $\Lambda$, define two
extremely different scales. The models under consideration are consistent with
the known experimental facts if $m_\Phi > 10^{-3} eV$ and
$\Lambda=\Lambda^{obs}\sim 10^{-56} cm^{-2}$. 2) Einstein week equivalence
principle is strictly satisfied and extended to scalar-tensor theories of
gravity using a novel form of principle of "constancy of fundamental
constants". 3) The dilaton plays simultaneously role of inflation field and
quintessence field and yields a sequential hyper-inflation with graceful exit
to asymptotic de Sitter space-time which is an attractor, and is approached as
$\exp(-\sqrt{3\Lambda^{obs}} ct/2)$. The time duration of inflation is $\Delta
t_{infl} \sim m_\Phi^{-1}$. 4) Ultra-high frequency ($\omega_\Phi \sim m_\Phi$)
dilatonic oscillations take place in asymptotic regime. 5) No fine tuning. (The
Robertson-Walker solutions of general type have the above properties.) 6) A
novel adjustment mechanism for cosmological constant problem seems to be
possible: the huge value of cosmological constant in the stringy frame is
re-scaled to its observed value by dilaton after transition to phenomenological
frame.
