We propose `Tapestry', a novel approach to pooled testing with application to COVID-19 testing with quantitative Reverse Transcription Polymerase Chain Reaction (RT-PCR) that can result in shorter testing time and conservation of reagents and testing kits. Tapestry combines ideas from compressed sensing and combinatorial group testing with a novel noise model for RT-PCR used for generation of synthetic data. Unlike Boolean group testing algorithms, the input is a quantitative readout from each test and the output is a list of viral loads for each sample relative to the pool with the highest viral load. While other pooling techniques require a second confirmatory assay, Tapestry obtains individual sample-level results in a single round of testing, at clinically acceptable false positive or false negative rates. We also propose designs for pooling matrices that facilitate good prediction of the infected samples while remaining practically viable. When testing $n$ samples out of which $k \ll n$ are infected, our method needs only $O(k \log n)$ tests when using random binary pooling matrices, with high probability. However, we also use deterministic binary pooling matrices based on combinatorial design ideas of Kirkman Triple Systems to balance between good reconstruction properties and matrix sparsity for ease of pooling. In practice, we have observed the need for fewer tests with such matrices than with random pooling matrices. This makes Tapestry capable of very large savings at low prevalence rates, while simultaneously remaining viable even at prevalence rates as high as 9.5\%. Empirically we find that single-round Tapestry pooling improves over two-round Dorfman pooling by almost a factor of 2 in the number of tests required. We validate Tapestry in simulations and wet lab experiments with oligomers in quantitative RT-PCR assays. Lastly, we describe use-case scenarios for deployment.