Distributed Compressive Sensing (DCS) is an extension of compressive sensing from single measurement vector to multiple signal vectors. The theory rests on the concept of joint sparsity of a signal ensemble which enables new algorithms that exploit both intra and inter-signal correlation. We consider DCS in the context of joint sparse model 1 (JSM-1). JSM-1 consists of an ensemble of correlated signals that can be decomposed into a sparse common component and a sparse innovation component. This paper proposes a new DCS recovery method for JSM-1. We recast JSM-1 as a sum of a sparse and low rank matrix and develop an algorithm to recover the signal from the compressed measurements using a rank-sparsity decomposition. The proposed method requires less number of measurements and provides better recovery accuracy compared to the existing algorithms. © 2019 IEEE.