At present, eDNA metabarcoding studies struggle to link genetic observations to underlying biology in a quantitative way, in large part due to biases during the PCR amplification process. Here we link previously disparate sets of techniques for making such data quantitative, showing that the underlying PCR mechanism explains observed patterns of amplicon data in a general way. By modeling the process through which amplicon-sequence data arises, rather than transforming the data post-hoc, we can estimate the starting proportions of DNA for many taxa simultaneously. This model can be calibrated with a variety of methods, including mock communities and variable-PCR-cycle sequencing runs, which we illustrate using simulations and in a series of empirical examples. Our approach opens the door to a wide range of applications in ecology, public health, and related fields.