The ambiguity of a nondeterministic finite automaton (NFA) $N$ for input size $n$ is the maximal number of accepting computations of $N$ for an input of size $n$. For all $k,r \in \mathbb{N}$ we construct languages $L_{r,k}$ which can be recognized by NFA's with size $k \cdot$poly$(r)$ and ambiguity $O(n^k)$, but $L_{r,k}$ has only NFA's with exponential size, if ambiguity $o(n^k)$ is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, 1989, Leung, 1998).