This paper proposes a new compressive sensing-based downlink channel state information (CSI) estimation scheme for frequency division duplexing (FDD) massive multiple-input multiple-output (MIMO) systems. The proposed scheme, which involves two-stage weighted block ℓ1-minimization, exploits the block sparse nature of the angular domain representation of the MIMO channel matrices and the existence of common scattering paths in the realistic propagation environment. In the first stage of the implemented scheme, a conventional block ℓ1-minimization program is solved to extract the information about the common/individual supports of the multiuser channel matrices. In the second stage, a weighted block ℓ1-minimization algorithm, the weighting coefficients of which are suitably chosen to exploit the acquired support information, is used to estimate the channel matrices. The analytic performance guarantees of the proposed scheme are specified based on the block restricted isometry property of the sensing matrix. Specifically, the upper bounds of the ℓ 2-norm reconstruction error are derived using various assumptions regarding the weighting. The obtained analytical results enable a discussion of the selection of weighting coefficients to enhance the CSI estimation performance, and the determination of a sufficient condition under which the proposed scheme outperforms the unweighted naive solution. Computer simulations show that the proposed method achieves higher estimation accuracy as compared to an existing greedy-based algorithm.