  Computing the renormalized masses and S-matrix elements in string theory,
involving states whose masses are not protected from quantum corrections,
requires defining off-shell amplitude with certain factorization properties.
While in the bosonic string theory one can in principle construct such an
amplitude from string field theory, there is no fully consistent field theory
for type II and heterotic string theory. In this paper we give a practical
construction of off-shell amplitudes satisfying the desired factorization
property using the formalism of picture changing operators. We describe a
systematic procedure for dealing with the spurious singularities of the
integration measure that we encounter in superstring perturbation theory. This
procedure is also useful for computing on-shell amplitudes, as we demonstrate
by computing the effect of Fayet-Iliopoulos D-terms in four dimensional
heterotic string theory compactifications using this formalism.
