The identification of genomic rearrangements, particularly in cancers, with high sensitivity and specificity using massively parallel sequencing remains a major challenge. Here, we describe the Genome Rearrangement IDentification Software Suite (GRIDSS), a high-speed structural variant (SV) caller that performs efficient genome-wide break-end assembly prior to variant calling using a novel positional de Bruijn graph assembler. By combining assembly, split read and read pair evidence using a probabilistic scoring, GRIDSS achieves high sensitivity and specificity on simulated, cell line and patient tumour data, recently winning SV sub-challenge #5 of the ICGC-TCGA DREAM Somatic Mutation Calling Challenge. On human cell line data, GRIDSS halves the false discovery rate compared to other recent methods. GRIDSS identifies non-template sequence insertions, micro-homologies and large imperfect homologies, and supports multi-sample analysis. GRIDSS is freely available at https://github.com/PapenfussLab/gridss.