We embed a small Josephson junction in a microwave resonator that allows simultaneous dc biasing and dispersive readout. Thermal fluctuations drive the junction into phase diffusion and induce a temperature-dependent shift in the resonance frequency. By sensing the thermal noise of a remote resistor in this manner, we demonstrate primary thermometry in the range of 300 mK to below 100 mK, and high-bandwidth (7.5 MHz) operation with a noise-equivalent temperature of better than 10 mu K/root Hz. At a finite bias voltage close to a Fiske resonance, amplification of the microwave probe signal is observed. We develop an accurate theoretical model of our device based on the theory of dynamical Coulomb blockade.