\indent Starting with $n$ particles at the origin in $Z^d$, let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a ball. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the ball, answering a question posed by Lawler in 1995 and confirming a prediction made by chemical physicists in the 1980's. Joint work with David Jerison and Scott Sheffield.