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The computationally prohibitive multi-dimensional searching procedure greatly restricts the application of the maximum likelihood (ML) direction-of-arrival (DOA) estimation method in practical systems. In this paper, we propose an efficient ML DOA estimator based on a spatially overcomplete array output formulation. The new method first reconstructs the array output on a predefined spatial discrete grid under the sparsity constraint via sparse Bayesian learning (SBL), thus obtaining a spatial power spectrum estimate that also indicates the coarse locations of the sources. Then a refined 1-D searching procedure is introduced to estimate the signal directions one by one based on the reconstruction result. The new method is able to estimate the incident signal number simultaneously. Numerical results show that the proposed method surpasses state-of-the-art methods largely in performance, especially in demanding scenarios such as low signal-to-noise ratio (SNR), limited snapshots and spatially adjacent signals.