We study the total branch length L_n of the Bolthausen–Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of L_n are presented. It is shown that L_n/E(L_n) converges to 1 in probability and that L_n, properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are applied to derive a corresponding limiting law for the total number of mutations for the Bolthausen–Sznitman coalescent with mutation rate r>0. Moreover, the results show that, for the Bolthausen–Sznitman coalescent, the total branch length L_n is closely related to X_n, the number of collision events that take place until there is just a single block. The proofs are mainly based on an analysis of random recursive equations using associated generating functions.