Metagenomics has opened new windows on investigating viral diversity and functions. Viromic studies typically require large sample volumes and filtration through 0.2 μm pore-size filters, consequently excluding or under-sampling tailed and very large viruses. We have optimized a targeted viromic approach that employs fluorescence-activated sorting and whole genome amplification to produce dsDNA-enriched libraries from discrete viral populations from a 1-ml water sample. Using this approach on an environmental sample from the Patagonian Shelf, we produced three distinct libraries. One of the virus libraries was dominated (79.65% of sequences with known viral homology) by giant viruses from the Mimiviridae and Phycodnaviridae families, while the two other viromes were dominated by smaller phycodnaviruses, cyanophages and other bacteriophages. The estimated genotypic richness and diversity in our sorted viromes, with 52-163 estimated genotypes, was much lower than in previous virome reports. Fragment recruitment of metagenome reads to selected reference viral genomes yields high genome coverage, suggesting little amplification and sequencing bias against some genomic regions. These results underscore the value of our approach as an effective way to target and investigate specific virus groups. In particular, it will help reveal the diversity and abundance of giant viruses in marine ecosystems.