Function fields of class number one

In 1975, J. Leitzel, M. Madan and C. Queen listed 7 function fields over finite fields (up to isomorphism) with positive genus and class number one. They claimed to prove that these were the only ones such. Recently, Claudio Stirpe found an 8th one! In this paper, we fix the argument in the former paper to show that this 8th example could have been found by their method and is the only one, so that the list is now complete.