The problem of full pixel target detection (FPTD) in hyperspectral images, assuming that the target spectral signature is known, is considered. We use the replacement target model (RTM). It exploits the fact that in FPTD problems the target completely fills the image pixel obscuring or replacing the background. The derivation of an RTM-based detection strategy is one of the original contributions of this work. It is assumed that both the background and the target classes are characterised by the Gaussian model and both share the same covariance matrix. Assuming that both the background mean vector and the covariance matrix are unknown, the fully adaptive detector (FAD) is derived. It is shown that the performance of the FAD in typical operating conditions can be approximated by those of the adaptive detector (AD) derived by assuming that the background mean vector is known. The AD is theoretically analysed, its performances are derived and its CFAR behaviour is demonstrated. It is also stated that, in practice, the AD design methodology can be adopted to design FAD automatic target detectors. This issue is proved by simulation in a case study in which the model parameters are estimated from a hyperspectral dataset acquired by the airborne visible/infrared imaging spectrometer (AVIRIS).