An exact theory is developed to describe the evolution of fibre fragmentation in a single-filament composite test as a function of the underlying fibre statistical strength and fibre/matrix interfacial shear stress, tau. The fragment distribution is a complicated function of fibre strength and tau because the stress around breaks which do occur recovers to the applied value, sigma, over a length delta(sigma) determined by tau. Therefore, no other breaks can occur within delta(sigma) of an existing break. To account for this effect, the fibre fragment distribution is decomposed into two parts; fragments formed by breaks separated by more than delta(sigma) at stress sigma, and fragments smaller than delta(sigma) which were formed at some prior stress sigma' textless sigma when a smaller delta(sigma') textless delta(sigma) prevailed. The distribution of fragments larger than delta(sigma) is identical to that of a fibre with a unique non-statistical strength sigma and is known exactly. The distribution of fragments smaller than delta(sigma) can then be determined from the distribution of the longer fragments. Predictions of the theory are compared to simulations of fibre fragmentation for several common models of stress recovery around fibre breaks with excellent agreement obtained. The present theory can be utilized to thus derive both the in situ fibre strength at short gauge lengths congruent-to delta and the tau from experimentally obtained fragment distributions, and an unambiguous inversion procedure is briefly discussed. The application of the theory to other muliple-cracking phenomena in composites is also discussed.