  Unambiguous B\"uchi automata, i.e. B\"uchi automata allowing only one
accepting run per word, are a useful restriction of B\"uchi automata that is
well-suited for probabilistic model-checking. In this paper we propose a more
permissive variant, namely finitely ambiguous B\"uchi automata, a
generalisation where each word has at most $k$ accepting runs, for some fixed
$k$. We adapt existing notions and results concerning finite and bounded
ambiguity of finite automata to the setting of $\omega$-languages and present a
translation from arbitrary nondeterministic B\"uchi automata with $n$ states to
finitely ambiguous automata with at most $3^n$ states and at most $n$ accepting
runs per word.
