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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. Terminating a coplanar waveguide 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 transition-edge sensor array readout. We derive an analytical model to reproduce the measurements over a wide range of bias currents, detunings and input powers. Tuning the frequency of the cavity by more than \SI{100}{\mega\hertz} with DC current, our device achieves a minimum current sensitivity of \SI{8.9}{\pico\ampere\per\sqrt{\hertz}}. Extrapolating the results of our analytical model, we predict an improved device based on our platform, capable of achieving sensitivities down to \SI{50}{\femto\ampere\per\sqrt{\hertz}}}, or even lower if one could 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.