We present a novel sparse signal reconstruction method "ISD", aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical L1 minimization approach. ISD addresses failed reconstructions of L1 minimization due to insufficient measurements. It estimates a support set I from a current reconstruction and obtains a new reconstruction by solving the minimization problem min{sum_{i not in I} |x_i| : Ax = b}, and it iterates these two steps for a small number of times. ISD differs from the orthogonal matching pursuit (OMP) method, as well as its variants, because (i) the index set I in ISD is not necessarily nested or increasing and (ii) the minimization problem above updates all the components of x at the same time. We generalize the Null Space Property to Truncated Null Space Property and present our analysis of ISD based on the latter. We introduce an efficient implementation of ISD, called threshold-ISD, for recovering signals with fast decaying distributions of nonzeros from compressive sensing measurements. Numerical experiments show that threshold-ISD has significant advantages over the classical L1 minimization approach, as well as two state-of-the-art algorithms: the iterative reweighted L1 minimization algorithm (IRL1) and the iterative reweighted least-squares algorithm (IRLS). MATLAB code is available for download from: http://www.caam.rice.edu/~optimization/L1/ISD/.