  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.
