This study presents a new fault diagnosis (FD) and fault-tolerant control (FTC) algorithm for a discrete-time non-Gaussian singular stochastic distribution control (SDC) system. An adaptive observer is used to diagnose the fault of the singular stochastic distribution system. Furthermore, the gain of the observer and adaptive updated law can be formulated by solving the corresponding linear matrix inequality. Based on the estimated fault information, the controller is reconfigured by minimising the performance index with regard to the entropy subjected to mean constraint. The reconfigured controller can make the output of the post-fault SDC system still have the minimum uncertainty, leading to the minimum entropy FTC of the non-Gaussian singular SDC system. Computer simulations are given to demonstrate the validity of the FD and minimum entropy FTC algorithms.