We address the self-stabilizing exact majority problem in the population protocol model, introduced by Angluin, Aspnes, Diamadi, Fischer, and Peralta (2004). In this model, there are \( n \) state machines, called agents, which form a network. At each time step, only two agents interact with each other, and update their states. In the self-stabilizing exact majority problem, each agent has a fixed opinion, \( \texttt{A}\) or \( \texttt{B}\) , and stabilizes to a safe configuration in which all agents output the majority opinion from any initial configuration. In this paper, we show the impossibility of solving the self-stabilizing exact majority problem without knowledge of \( n \) in any protocol. We propose a silent self-stabilizing exact majority protocol, which stabilizes within \( O(n) \) parallel time in expectation and within \( O(n \log n) \) parallel time with high probability, using \( O(n) \) states, with knowledge of n. Here, a silent protocol means that, after stabilization, the state of each agent does not change. We establish lower bounds, proving that any silent protocol requires \( \varOmega (n) \) states, \( \varOmega (n) \) parallel time in expectation, and \( \varOmega (n \log n) \) parallel time with high probability to stabilize. Thus, the proposed protocol is time- and space-optimal.