  The critical coagulation-fragmentation equation with multiplicative
coagulation and constant fragmentation kernels is known to not have global
mass-conserving solutions when the initial mass is greater than $1$. We show
that for any given positive initial mass with finite second moment, there is a
time $T^*>0$ such that the equation possesses a unique mass-conserving solution
up to $T^*$. The novel idea is to singularly perturb the constant fragmentation
kernel by small additive terms and study the limiting behavior of the solutions
of the perturbed system via the Bernstein transform.
