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The detection of information-bearing Gaussian processes immersed in additive white Gaussian noise (WGN) is an important problem that arises in many signal processing applications. When the level of the WGN is unknown, classical approaches to the problem fail. In this paper a principle of invariance is used to derive a detector with performance that is invariant (or insensitive) to system gain, or equivalently channel attenuation. The detector structure can be realized and detection thresholds set without prior knowledge of the WGN level. When the observation interval is large the detector has the structure of a spectral estimator-correlator, the output of which is compared to an adaptive threshold. The invariance feature of the detector makes it a constant false alarm rate (CFAR) receiver; an ad hoc structure for suboptimal CFAR Gauss-Gauss detection is discussed as well.