Viruses are the most abundant biological entities on Earth and play key roles in host ecology, evolution, and horizontal gene transfer. Despite recent progress in viral metagenomics, the inherent genetic complexity of virus populations still poses technical difficulties for recovering complete virus genomes from natural assemblages. To address these challenges, we developed an assembly-free, single-molecule nanopore sequencing approach, enabling direct recovery of complete virus genome sequences from environmental samples. Our method yielded thousands of full-length, high-quality draft virus genome sequences that were not recovered using standard short-read assembly approaches. Additionally, our analyses discriminated between populations whose genomes had identical direct terminal repeats versus those with circularly permuted repeats at their termini, thus providing new insight into native virus reproduction and genome packaging. Novel DNA sequences were discovered, whose repeat structures, gene contents, and concatemer lengths suggest they are phage-inducible chromosomal islands, which are packaged as concatemers in phage particles, with lengths that match the size ranges of co-occurring phage genomes. Our new virus sequencing strategy can provide previously unavailable information about the genome structures, population biology, and ecology of naturally occurring viruses and viral parasites.