We compare new maps of the hot gas, dark matter, and galaxies for 1E 0657- 56, a cluster with a rare high-velocity merger occurring nearly in the plane of the sky. The X-ray observations reveal a bullet-like gas subcluster just exiting the collision site. A prominent bow shock gives an estimate of the subcluster velocity, 4500 km s super(-1), which lies mostly in the plane of the sky. The optical image shows that the gas lags behind the subcluster galaxies. The weak- lensing mass map reveals a dark matter clump lying ahead of the collisional gas bullet but coincident with the effectively collisionless galaxies. From these observations, one can directly estimate the cross section of the dark matter self-interaction. That the dark matter is not fluid-like is seen directly in the X-ray-lensing mass overlay; more quantitative limits can be derived from three simple independent arguments. The most sensitive constraint, capital sigma /m < 1 cm super(2) g super(-1), comes from the consistency of the subcluster mass-to-light ratio with the main cluster (and universal) value, which rules out a significant mass loss due to dark matter particle collisions. This limit excludes most of the 0.5-5 cm super(2) g super(-1) interval proposed to explain the flat mass profiles in galaxies. Our result is only an order-of-magnitude estimate that involves a number of simplifying, but always conservative, assumptions; stronger constraints may be derived using hydrodynamic simulations of this cluster.