Frequent and widespread testing of members of the population who are asymptomatic for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is essential for the mitigation of the transmission of the virus. Despite the recent increases in testing capacity, tests based on quantitative polymerase chain reaction (qPCR) assays cannot be easily deployed at the scale required for population-wide screening. Here, we show that next-generation sequencing of pooled samples tagged with sample-specific molecular barcodes enables the testing of thousands of nasal or saliva samples for SARS-CoV-2 RNA in a single run without the need for RNA extraction. The assay, which we named SwabSeq, incorporates a synthetic RNA standard that facilitates end-point quantification and the calling of true negatives, and that reduces the requirements for automation, purification and sample-to-sample normalization. We used SwabSeq to perform 80,000 tests, with an analytical sensitivity and specificity comparable to or better than traditional qPCR tests, in less than two months with turnaround times of less than 24 h. SwabSeq could be rapidly adapted for the detection of other pathogens.