Sparse convex optimization involves optimization problems where the decision variables are constrained to have a certain number of entries equal to zero. In this paper, we consider the case in which the objective function is decomposed into a sum of different local objective functions and propose a novel fully-distributed scheme to address the problem over a network of cooperating agents. Specifically, by taking advantage of a suitable problem reformulation, we define an Augmented Lagrangian function associated with the reformulated problem. Then, we address such an Augmented Lagrangian by suitably interlacing the Gradient Tracking distributed algorithm and the Block Coordinated Descent method giving rise to a novel fully-distributed scheme. The effectiveness of the proposed algorithm is corroborated through some numerical simulations of problems considering both synthetic and real-world data sets.