  Exact solutions to the quantum S-matrices for solitons in simply-laced affine
Toda field theories are obtained, except for certain factors of simple type
which remain undetermined in some cases. These are found by postulating
solutions which are consistent with the semi-classical limit, $\hbar\rightarrow
0$, and the known time delays for a classical two soliton interaction. This is
done by a `$q$-deformation' procedure, to move from the classical time delay to
the exact S-matrix, by inserting a special function called the `regularised'
quantum dilogarithm, which only holds when $|q|=1$. It is then checked that the
solutions satisfy the crossing, unitarity and bootstrap constraints of S-matrix
theory. These properties essentially follow from analogous properties satisfied
by the classical time delay. Furthermore, the lowest mass breather S-matrices
are computed by the bootstrap, and it is shown that these agree with the
particle S-matrices known already in the affine Toda field theories, in all
simply-laced cases.
