We propose a mesoscopic kinetic-inductance radiation detector based on a long superconductor-normal metal-superconductor Josephson junction. The operation of this proximity Josephson sensor relies on large kinetic inductance variations under irradiation due to the exponential temperature dependence of the critical current. Coupled with a dc superconducting quantum interference device readout, the PJS is able to provide a signal to noise (S/N) ratio up to ∼103 in the terahertz regime if operated as calorimeter, while electrical noise equivalent power as low as ∼7×10−20W∕Hz at 200mK can be achieved in the bolometer operation. The high performance together with the ease of fabrication make this structure attractive as an ultrasensitive cryogenic detector of terahertz electromagnetic radiation.