  We demonstrate the existence of a non-terminating game of Beggar-My-Neighbor,
discovered by lead author Brayden Casella. We detail the method for
constructing this game and identify a cyclical structure of 62 tricks that is
reached by 30 distinct starting hands. We further present a short history of
the search for this solution since the problem was posed, and a record of
previously found longest terminating games. The existence of this
non-terminating game provides a solution to a long-standing question which John
H. Conway called an `anti-Hilbert problem.'
