Here we propose an accelerated version of one of the most promising methods for audio source separation proposed by Duong et al. [“Under-determined reverberant audio source separation using a full-rank spatial covariance model,” IEEE Trans. ASLP, vol. 18, no. 7, pp. 1830-1840, Sep. 2010]. We refer to this conventional method as full-rank spatial covariance analysis (FCA), and the proposed method as FastFCA. A major drawback of the conventional FCA is computational complexity: inversion and multiplication of covariance matrices are required at each time-frequency point and each EM iteration. To overcome this drawback, the proposed FastFCA diagonalizes the covariance matrices jointly based on the generalized eigenvalue problem. This leads to significantly reduced computational complexity of the FastFCA, because the complexity of matrix inversion and matrix multiplication for diagonal matrices is O(M) instead of O(M³) (M: matrix order). Furthermore, the FastFCA is rigorously equivalent to the FCA, and therefore the reduction in computational complexity is realized without degradation in source separation performance. An experiment showed that the FastFCA was over 250 times faster than the FCA with virtually no degradation in source separation performance. In this paper, we focus on the two-source case, while the case of more than two sources is treated in a separate paper.