At fast timescales, the self-similarity of random Brownian motion is expected to break down and be replaced by ballistic motion. So far, an experimental verification of this prediction has been out of reach due to a lack of instrumentation fast and precise enough to capture this motion. With a newly developed detector, we have been able to observe the Brownian motion of a single particle in an optical trap with 75 MHz bandwidth and sub-{AA}ngstrom spatial precision. We report the first measurements of ballistic Brownian motion as well as the first determination of the velocity autocorrelation function of a Brownian particle. The data are in excellent agreement with theoretical predictions taking into account the inertia of the particle and the surrounding fluid as well as hydrodynamic memory effects.