This paper presents a closed-form noniterative transformation of the Maxwell-Garnett mixing rule for biphased mixtures to the triple-pole Debye relaxation formula. For the first time, it is formally proven that such a transformation is complete for conductive constituent materials. In other words, the Maxwell-Garnett representation of any biphased mixture of any conductive materials always has its formal equivalent in the Debye form with three poles at most. For specific aspect ratios of ellipsoidal inclusions, the number of poles reduces to one or two, which is formally proven herein, while in previous studies, a single-pole Debye model was arbitrarily assumed. The proposed transformation provides Debye parameters as an explicit function of a mixture composition, which is competitive to alternative techniques based on laborious curve-fitting algorithms. The newly proposed approach is of particular importance to time-domain modeling of dilute mixtures, where the Maxwell-Garnett mixing rule is usually approximated with available dispersive models. Computational examples given in this paper show advantages of the presented method over previous Maxwell-Garnett to Debye conversion algorithms, in terms of accuracy, robustness, and computational cost.