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This paper presents the proof of an Optimum mixture estimator for the single channel speech separation problem, which is a technique for separating two speech signals from a single recording of their mixture. The presented work is an attempt to solve a fundamental limitation in the current single channel speech separation techniques, in which it is assumed that the data used in the training as well as test phases of the separation model have the same energy levels. To overcome this limitation, a gain adapted Optimum mixture estimator is derived, which estimates the mixture of speech signals under the different signal-to-signal ratios (SSRs). Specifically, the speakers' gains are incorporated as unknown parameters into the separation model, and then the estimator is derived in terms of the source distributions and SSR. It is demonstrated that the use of the Optimum mixture estimator results in the lower estimation error than the non-linear mapping (log and inverse-log operations)-based Mixture-Maximization (MixMax) or Quadratic estimators. The experimental results based on the real speech data also depict that the proposed estimator improves the mixture estimation performance significantly when compared with MixMax or Quadratic estimators with the gain adaptation. [ABSTRACT FROM AUTHOR]