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 8.9pA/Hz. Extrapolating the results of our analytical model, we predict an improved device based on our platform, capable of achieving a sensitivity down to 50fA/Hz, or even lower if one can take advantage of parametric amplification in the Josephson cavity. Taking advantage of the Josephson architecture, our approach can provide higher sensitivity than kinetic inductance designs, and potentially enables detection of currents ultimately limited by quantum noise.