This paper deals with Low-Density Construction-A (LDA) lattices, which are obtained via Construction A from non-binary low-density parity-check codes. More precisely, a proof is provided that Voronoi constellations of LDA lattices achieve capacity of the AWGN channel under lattice encoding and decoding for every signal-to-noise ratio greater than 1. This is obtained after showing the same result for more general Construction-A lattice constellations. The theoretical analysis is carried out in a way that allows to describe how the prime number underlying Construction A behaves as a function of the lattice dimension. Moreover, no dithering is required in the transmission scheme, simplifying some previous solutions of the problem. Remarkably, capacity is achievable with LDA lattice codes whose parity-check matrices have constant row and column Hamming weights. Some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound for the minimum Euclidean distance of LDA lattices in our ensemble. © 1963-2012 IEEE.