By solving exactly the Faddeev equations for the bound-state problem of three mesons, we demonstrate that current theoretical predictions pointing to the existence of a deeply-bound doubly bottom axial vector tetraquark lead to the existence of a unique bound state of three B mesons. We find that the BB^*B^*-B^*B^*B^* state with quantum numbers (I)J^P=(1/2)2^-, T_bbb, is about 90 MeV below any possible three B-meson threshold for the reported binding of the doubly bottom axial vector tetraquark, T_bb.