  For solving the $2\to 2,3$ three-body Coulomb scattering problem the
Faddeev-Merkuriev integral equations in discrete Hilbert-space basis
representation are considered. It is shown that as far as scattering amplitudes
are considered the error caused by truncating the basis can be made arbitrarily
small. By this truncation also the Coulomb Green's operator is confined onto
the two-body sector of the three-body configuration space and in leading order
can be constructed with the help of convolution integrals of two-body Green's
operators. For performing the convolution integral an integration contour is
proposed that is valid for all energies, including bound-state as well as
scattering energies below and above the three-body breakup threshold.
