This paper proposes a new compressive sensing based downlink channel state information (CSI) estimation scheme for FDD massive MIMO systems. The proposed approach, which involves two-stage weighted block ℓ1-minimization, exploits the block sparse nature of the angular domain representation of the MIMO channel matrices, as well as the existence of common scattering paths in the realistic propagation environment. In the first stage of our method, a conventional block ℓ1-minimization program is solved to extract the information about the common/individual supports of the multi-user channel matrices. In the second stage, a weighted block ℓ1-minimization algorithm, with the weighting coefficients suitably chosen to exploit the acquired support knowledge, is then performed for channel matrix estimation. Analytic performance guarantees of the proposed method are specified using the block restricted isometry property of the sensing matrix; specifically, the I-norm reconstruction error upper bounds achieved by our approach are derived. The analytic results allow us to discuss the selection of weighting coefficients for enhancing CSI estimation performance. Computer simulations show that our method achieves better estimation accuracy as compared to an existing greedy-based algorithm.