  A precise evaluation of the two-loop vacuum energy is provided for certain
Z_2 x Z_2 Calabi-Yau orbifold models in the Heterotic string. The evaluation is
based on the recent general prescription for superstring perturbation theory in
terms of integration over cycles in supermoduli space, implemented at two-loops
with the gauge-fixing methods based on the natural projection of supermoduli
space onto moduli space using the super-period matrix. It is shown that the
contribution from the interior of supermoduli space (computed with the
procedure that has been used in previous two-loop computations) vanishes
identically for both the E_8 x E_8 and Spin (32)/Z_2 Heterotic strings. The
contribution from the boundary of supermoduli space is also evaluated, and
shown to vanish for the E_8 x E_8 string but not for the Spin (32)/Z_2 string,
thus breaking supersymmetry in this last model. As a byproduct, the vacuum
energy in Type II superstrings is shown to vanish as well for these orbifolds.
