  We have embedded 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 from
300 mK to below 100 mK, and high-bandwidth (7.5 MHz) operation with a
noise-equivalent temperature of better than 10 $\mathrm{\mu K/\sqrt{Hz}}$. 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.
