The Telomere-to-Telomere consortium recently assembled the first truly complete sequence of a human genome. To resolve the most complex repeats, this project relied on manual integration of ultra-long Oxford Nanopore sequencing reads with a high-resolution assembly graph built from long, accurate PacBio HiFi reads. We have improved and automated this strategy in Verkko, an iterative, graph-based pipeline for assembling complete, diploid genomes. Verkko begins with a multiplex de Bruijn graph built from long, accurate reads and progressively simplifies this graph via the integration of ultra-long reads and haplotype-specific markers. The result is a phased, diploid assembly of both haplotypes, with many chromosomes automatically assembled from telomere to telomere. Running Verkko on the HG002 human genome resulted in 20 of 46 diploid chromosomes assembled without gaps at 99.9997% accuracy. The complete assembly of diploid genomes is a critical step towards the construction of comprehensive pangenome databases and chromosome-scale comparative genomics.