Separation of underdetermined speech mixtures, where the number of speakers is greater than the number of microphones, is a challenging task. Due to the intermittent behaviour of human conversations, typically, the instantaneous number of active speakers does not exceed the number of microphones, namely the mixture is locally (over-)determined. This scenario is addressed in this paper using a dual stage approach: diarization followed by separation. The diarization stage is based on spectral decomposition of the correlation matrix between different time frames. Specifically, the spectral gap reveals the overall number of speakers, and the computed eigenvectors form a simplex of the activity of the speakers across time. In the separation stage, the diarization results are utilized for estimating the mixing acoustic channels, as well as for constructing an unmixing scheme for extracting the individual speakers. The performance is demonstrated in a challenging scenario with six speakers and only four microphones. The proposed method shows perfect recovery of the overall number of speakers, close to perfect diarization accuracy, and high separation capabilities in various reverberation conditions.