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We present the design, measurement, and analysis of a current sensor based on a process of Josephson parametric upconversion in a superconducting microwave cavity. When a coplanar waveguide is terminated with a nanobridge-constriction Josephson junction, we observe modulation sidebands from the cavity that enable highly sensitive frequency-multiplexed output of small currents for applications such as readout of transition-edge sensor arrays. We derive an analytical model to reproduce the measurements over a wide range of bias current, detuning, and input power. When the frequency of the cavity is tuned by more than 100 MHz with a dc current, our device achieves a minimum current sensitivity of . Extrapolating the results of our analytical model, we predict an improved device based on our platform, capable of achieving a sensitivity down to , or even lower if one can take …