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Solutions of the synthetic clay Laponite are strongly viscoelastic, even at very low particle concentrations. The formation of a gel, evidenced by the existence of a fractal network, has been invoked in explaining the viscoelasticity. We study the structure and viscosity of Laponite using static light scattering and rheometry. Contrary to previous observations, we find no evidence of a fractal-like organization of the colloidal particles, provided the dispersion is prepared carefully. The results show that there is no relation between the apparent fractal dimension and the viscoelasticity. A possible interpretation of our results is that Laponite solutions form colloidal glasses, rather than gels.