  We define a multiple Dirichlet series whose group of functional equations is
the Weyl group of the affine Kac-Moody root system $\tilde{A}_n$, generalizing
the theory of multiple Dirichlet series for finite Weyl groups. The
construction is over the rational function field $\mathbb{F}_q(t)$, and is
based upon four natural axioms from algebraic geometry. We prove that the four
axioms yield a unique series with meromorphic continuation to the largest
possible domain and the desired infinite group of symmetries.
