We focus on the problem of detecting a signal in compound-Gaussian clutter, where the texture is a random variable with Gamma or inverse Gamma distribution. The persymmetric structure of the covariance matrix is exploited and a persymmetric generalized likelihood ratio test (Per-GLRT) using a three-step procedure is proposed. In addition, we prove that the Per-GLRT ensures constant false alarm rate (CFAR) property with respect to the covariance matrix. Finally, the detector is assessed by Monte Carlo simulations. Performance comparison of the Per-GLRT with the traditional GLRT shows that the former improves the detection performance in training-limited scenarios.