A single-channel adaptive algorithm for a detection of a fast fluctuating Gaussian random disturbance with the unknown appearance time and dispersion is synthesized. Using the method of local-Markov approximation of the decision determining statistics, or its increment, the closed analytical expressions for the false alarm and missing probabilities are found. By the methods of statistical simulation, it is established that the proposed detector is operable, and the analytical formulas describing the quality of its functioning are in good agreement with the corresponding experimental data in a wide range of values of the analyzed random process parameters.