A double-periodic array of pairs of parallel gold nanorods is shown to have a negative refractive index in the optical range. Such behavior results from the plasmon resonance in the pairs of nanorods for both the electric and the magnetic components of light. The refractive index is retrieved from direct phase and amplitude measurements for transmission and reflection, which are all in excellent agreement with simulations. Both experiments and simulations demonstrate that a negative refractive index n' approximately -0.3 is achieved at the optical communication wavelength of 1.5 microm using the array of nanorods. The retrieved refractive index critically depends on the phase of the transmitted wave, which emphasizes the importance of phase measurements in finding n'.