One of the main challenges in containing the Coronoavirus disease 2019 (COVID-19) pandemic stems from the difficulty in carrying out efficient mass diagnosis over large populations. The leading method to test for COVID-19 infection utilizes qualitative polymerase chain reaction, implemented using dedicated machinery which can simultaneously process a limited amount of samples. A candidate method to increase the test throughput is to examine pooled samples comprised of a mixture of samples from different patients. In this work we study pooling-based COVID-19 tests. We identify the specific requirements of COVID-19 testing, including the need to characterize the infection level and to operate in a one-shot fashion, which limit the application of traditional group-testing (GT) methods. We then propose a multi-level GT scheme, designed specifically to meet the unique requirements of COVID-19 tests, while exploiting the strength of GT theory to enable accurate recovery using much fewer tests than patients. Our numerical results demonstrate that multi-level GT reliably and efficiently detects the infection levels, while achieving improved accuracy over previously proposed one-shot COVID-19 pooled-testing methods.