Singleton-language test for finite automata

To decide whether a finite-state automaton accepts exactly one distinct word, first obtain some accepted word $w$ by a finite automaton emptiness test. If there is none, answer no. Otherwise construct a finite-state automaton for the accepted formal language minus the singleton $\{w\}$ and test it for emptiness. Answer yes exactly when that difference is empty. Subset construction handles multiple accepting runs without confusing runs with distinct words.