Let $E = \{\ldots ,-2,-1, 0, 1, 2, \ldots \}$. Given a uni-instantaneous bilateral birth-death pre-generator matrix $Q = (q_{ij})$ defined on $E \times E$, we investigate the existence conditions for the corresponding birth-death $Q$-processes, and furthermore, construct all the processes when the existence conditions are satisfied. In addition, the uniqueness of the honest $Q$-processes are investigated as well.