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Recently it has been predicted that materials with exceptionally strong optical activity may also possess a negative refractive index, allowing the realization of superlenses for super-resolution imaging and data storage applications. Here we demonstrate experimentally and numerically that a chirality-induced negative index of refraction is possible. A negative index of refraction due to three-dimensional chirality is demonstrated for a bilayered metamaterial based on pairs of mutually twisted planar metal patterns in parallel planes, which also shows negative electric and magnetic responses and exceptionally strong optical activity and circular dichroism. Multilayered forms of the metamaterial are found to be suitable for use as ultrathin polarization rotators and circular polarizers for practical applications.