We investigate the limits of ground-based astrometry with adaptive optics using the core of the Galactic globular cluster M5. Adaptive optics systems provide near diffraction-limit imaging with the world's largest telescopes. The substantial improvement in both resolution and signal-to-noise ratio enables high-precision astrometry from the ground. We describe the dominant systematic errors that typically limit ground-based differential astrometry, and enumerate observational considerations for mitigating their effects. After implementing these measures, we find that the dominant limitation on astrometric performance in this experiment is caused by tilt anisoplanatism. We then present an optimal estimation technique for measuring the position of one star relative to a grid of reference stars in the face of this correlated random noise source. Our methodology has the advantage of reducing the astrometric errors as the square root of time and faster than the square root of the number of reference stars -- effectively eliminating noise caused by atmospheric tilt to the point that astrometric performance is limited by centering accuracy. Using 50 reference stars we demonstrate single-epoch astrometric precision of ~ 1 mas in 1 second, decreasing to < 100 microarcseconds in 2 minutes of integration time at the Hale 200-inch telescope. We also show that our astrometry is accurate to

32 pages, 12 figures; submitted to AJ