The concept of invariance in hypothesis testing is brought to bear on the problem of detecting signals of known form and unknown energy in Gaussian noise of unknown level. The noise covariance function is assumed to beK(t,u) = sigma^2 pho(t,u)whererho(t,u)is the known form of the covariance function andsigma^2is the unknown level. Classical approaches to signal detection depend on the assumption thatK(t,u)is known completely. Then, a correlation-type receiver that is the uniformly most powerful (UMP) test ofH_o(signal absent) versusH_1(signal present) can be derived. Whensigma^2is unknown, there exists no UMP test. However, it is shown in this paper that there exists a test ofH_oversusH_1that is UMP-invariant for a very natural group of transformations on the space of observations. The derived test is found to be independent of knowledge about the noise levelsigma^2, since the derived test (receiver) contains an error-free estimate ofsigma^2. This utopian conclusion is reconciled by noting that the derived receiver can never be physically realized. It is shown that any physically realizable version of the receiver has at-distributed test statistic. This permits choice of operating receiver thresholds and evaluation of performance characteristics.