  The N-point amplitudes for the Type II and Heterotic superstrings at two-loop
order and for $N \leq 4$ massless NS bosons are evaluated explicitly from first
principles, using the method of projection onto super period matrices
introduced and developed in the first five papers of this series. The
gauge-dependent corrections to the vertex operators, identified in paper V, are
carefully taken into account, and the crucial counterterms which are Dolbeault
exact in one insertion point and de Rham closed in the remaining points are
constructed explicitly. This procedure maintains gauge slice independence at
every stage of the evaluation.
  Analysis of the resulting amplitudes demonstrates, from first principles,
that for $N\leq 3$, no two-loop corrections occur, while for N=4, no two-loop
corrections to the low energy effective action occur for $R^4$ terms in the
Type II superstrings, and for $F^4$, $F^2F^2$, $F^2R^2$, and $R^4$ terms in the
Heterotic strings.
