  We describe the structure of all codimension-two lattice configurations $A$
which admit a stable rational $A$-hypergeometric function, that is a rational
function $F$ all whose partial derivatives are non zero, and which is a
solution of the $A$-hypergeometric system of partial differential equations
defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all
stable rational $A$-hypergeometric functions may be described by toric residues
and apply our results to study the rationality of bivariate series whose
coefficients are quotients of factorials of linear forms.
