We report on a characterization of the chaotic optical field from a high-gain, self-amplified spontaneous-emission (SASE) free-electron laser. The temporal structure of the amplitude and phase are measured in a single-shot mode, with a resolution well below the coherence length, and the statistics over multiple pulses is determined. The measurement is in excellent quantitative agreement with the prediction based on analysis of random noise, and further verifies the chaotic nature of the SASE optical field.