  In perturbative quantum field theory the limit of compactification on an
almost light-like circle has recently been shown to be plagued by divergences.
We argue that the light-like limit for M-theory probably is free of such
divergences due to, among others, the existence of the wrapping modes of the
membranes. To illustrate this, we consider superstring theory compactified on
an almost light-like circle. Specifically, we compute a one-loop four-point
amplitude in type II theory. As is well known, if the external states have
vanishing momenta in the compact dimension, the divergence in the light-like
limit is even stronger than in field theory. However, in the case of present
interest, where these external momenta are non-vanishing, there is a subtle
compensation and the resulting amplitude has a well-defined and finite
light-like limit. The net effect of taking the light-like limit is to replace
the integration over one of the moduli of the 4-punctured torus by a sum over a
discrete modulus taking values in a finite lattice on the torus. The same
result can also be obtained from a suitably "Wick rotated" amplitude computed
directly with a compact light-like circle.
