  The $Z_2\times Z_2$ heterotic string orbifold gives rise to a large space of
phenomenological three generation models that serves as a testing ground to
explore how the Standard Model of particle physics may be incorporated in a
theory of quantum gravity. Recently, we demonstrated the existence of type 0
$Z_2\times Z_2$ heterotic string orbifolds in which there are no massless
fermionic states. In this paper we demonstrate the existence of
non--supersymmetric tachyon--free $Z_2\times Z_2$ heterotic string orbifolds
that do not contain any massless bosonic states from the twisted sectors. We
dub these configurations type ${\bar 0}$ models. They necessarily contain
untwisted bosonic states, producing the gravitational, gauge and scalar moduli
degrees of freedom, but possess an excess of massless fermionic states over
bosonic ones, hence producing a positive cosmological constant. Such
configurations may be instrumental when trying to understand the string
dynamics in the early universe.
