We present broadband observations and spectral modeling of PKS B0008-421, and identify it as an extreme gigahertz-peaked spectrum (GPS) source. PKS B0008-421 is characterized by the steepest known spectral slope below the turnover, close to the theoretical limit of synchrotron self-absorption, and the smallest known spectral width of any GPS source. Spectral coverage of the source spans from 0.118 to 22 GHz, which includes data from the Murchison Widefield Array and the wide bandpass receivers on the Australia Telescope Compact Array. We have implemented a Bayesian inference model fitting routine to fit the data with various absorption models. We find that without the inclusion of a high-frequency exponential break the absorption models can not accurately fit the data, with significant deviations above and below the peak in the radio spectrum. The addition of a high-frequency break provides acceptable spectral fits for the inhomogeneous free-free absorption and double-component synchrotron self-absorption models, with the inhomogeneous free-free absorption model statistically favored. The requirement of a high-frequency spectral break implies that the source has ceased injecting fresh particles. Additional support for the inhomogeneous free-free absorption model as being responsible for the turnover in the spectrum is given by the consistency between the physical parameters derived from the model fit and the implications of the exponential spectral break, such as the necessity of the source being surrounded by a dense ambient medium to maintain the peak frequency near the gigahertz region. The discovery of PKS B0008-421 suggests that the next generation of low radio frequency surveys could reveal a large population of GPS sources that have ceased activity, and that a portion of the ultra-steep spectrum source population could be composed of these GPS sources in a relic phase.
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