This paper introduces a theoretically-rigorous sound source localization (SSL) method based on a robust extension of the classical multiple signal classification (MUSIC) algorithm. The original SSL method estimates the noise eigenvectors and the MUSIC spectrum by computing the spatial covariance matrix of the observed multichannel signal and then detects the peaks from the spectrum. In this work, the covariance matrix is replaced with the positive definite shape matrix originating from the elliptically contoured α-stable model, which is more suitable under real noisy high-reverberant conditions. Evaluation on synthetic data shows that the proposed method outperforms baseline methods under such adverse conditions, while it is comparable on real data recorded in a mild acoustic condition.