Start with n particles at the origin in Z^d, and let each particle in turn perform simple random walk until reaching an unoccupied site. This process, known as internal DLA, is a discrete analogue of the stable direction of Hele-Shaw flow. Lawler, Bramson and Griffeath proved that with high probability the random set of n occupied sites formed by internal DLA 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. These logarithmic fluctuations were predicted numerically by chemical physicists in the 1980s. Joint work with David Jerison and Scott Sheffield.