Using a high resolution SQUID voltmeter, we have measured the spectrum of low frequency voltage fluctuations across a thin-film Josephson tunnel junction biased at a constant current I greater than the junction critical current I c . We find that the frequency dependence of the voltage spectrum V^{2}(f) may be accurately represented by the power law V^{2}(f) \propto f^{-1} over the frequency range of our data: 10^{-2} Hz. The dependence of the magnitude of the spectra at any single frequency upon the value of the bias current I and upon the sample temperature T supports our hypothesis that the observed voltage fluctuations arise from a modulation of the junction critical current I c by equilibrium, thermodynamic temperature fluctuations in the active junction volume. We are able to interpret our measurements in terms of the semi-empirical theory of Clarke and Voss for the low frequency fluctuation spectrum of systems obeying a diffusion equation. This interpretation provides design criteria which may prove useful in reducing the level of long-term drifts in systems employing Josephson tunnel junctions.