In this paper we study the approximate minimization problem for language modelling. We assume we are given some language model as a black box. The objective is to obtain a weighted finite automaton (WFA) that fits within a given size constraint and which mimics the behaviour of the original model while minimizing some notion of distance between the black box and the extracted WFA. We provide an algorithm for the approximate minimization of black boxes trained for language modelling of sequential data over a one-letter alphabet. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. This allows us to use the spectral norm to measure the distance between the black box and the WFA. We provide theoretical guarantees to study the potentially infinite-rank Hankel matrix of the black box, without accessing the training data, and we prove that our method returns an asymptotically-optimal approximation.