  Existence of mass-conserving weak solutions to the coagulation-fragmentation
equation is established when the fragmentation mechanism produces an infinite
number of fragments after splitting. The coagulation kernel is assumed to
increase at most linearly for large sizes and no assumption is made on the
growth of the overall fragmentation rate for large sizes. However, they are
both required to vanish for small sizes at a rate which is prescribed by the
(non-integrable) singularity of the fragment distribution.
