Recovery of ribosomal small subunit genes by assembly of short read community DNA sequence data generally fails, making taxonomic characterization difficult. Here, we solve this problem with a novel iterative method, based on the expectation maximization algorithm, that reconstructs full-length small subunit gene sequences and provides estimates of relative taxon abundances. We apply the method to natural and simulated microbial communities, and correctly recover community structure from known and previously unreported rRNA gene sequences. An implementation of the method is freely available at https://github.com/csmiller/EMIRGE.