In this Letter, a compressed channel estimation approach referred to as burst weighted ℓ1-minimisation is proposed to alleviate the overwhelming pilot overhead in the downlink path of frequency division duplex (FDD) massive MIMO systems. The proposed approach exploits the burst-sparse property of channel as the prior support information (PSI) and sets adaptive weights for the channel coefficients according to the accuracy of PSI values. Furthermore, a novel greedy algorithm is elaborated, which extracts PSI from the massive MIMO channel. Simulation results demonstrate that the proposed algorithm outperforms other solutions in terms of the channel state information (CSI) estimation error.