  Gessel walks are lattice walks in the quarter plane $\set N^2$ which start at
the origin $(0,0)\in\set N^2$ and consist only of steps chosen from the set
$\{\leftarrow,\swarrow,\nearrow,\to\}$. We prove that if $g(n;i,j)$ denotes the
number of Gessel walks of length $n$ which end at the point $(i,j)\in\set N^2$,
then the trivariate generating series $G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^i
y^j t^n$ is an algebraic function.
