We show that any one-counter automaton with n states, if its language is non-empty, accepts some word of length at most O(n 2 ). This closes the gap between the previously known upper bound of O(n 3 ) and lower bound of Ω(n 2 ). More generally, we prove a tight upper bound on the length of shortest paths between arbitrary configurations in one-counter transition systems (weaker bounds have previously appeared in the literature).