  We present a computer-aided, yet fully rigorous, proof of Ira Gessel's
tantalizingly simply-stated conjecture that the number of ways of walking $2n$
steps in the region $x+y \geq 0, y \geq 0$ of the square-lattice with unit
steps in the east, west, north, and south directions, that start and end at the
origin, equals $16^n\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ .
