  The Abelian sandpile growth model is a diffusion process for configurations
of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which
sites with at least 2d chips {\em topple}, distributing 1 chip to each of their
neighbors in the lattice, until no more topplings are possible. From an initial
configuration consisting of $n$ chips placed at a single vertex, the rescaled
stable configuration seems to converge to a particular fractal pattern as $n\to
\infty$. However, little has been proved about the appearance of the stable
configurations. We use PDE techniques to prove that the rescaled stable
configurations do indeed converge to a unique limit as $n \to \infty$. We
characterize the limit as the Laplacian of the solution to an elliptic obstacle
problem.
