Relativistic viscous hydrodynamic fits to Brookhaven Relativistic Heavy Ion Collider data on the centrality dependence of multiplicity, transverse, and elliptic flow for square root s = 200 GeV Au+Au collisions are presented. For standard (Glauber-type) initial conditions, while data on the integrated elliptic flow coefficient v(2) are consistent with a ratio of viscosity over entropy density up to eta/s approximately 0.16, data on minimum bias v(2) seem to favor a much smaller viscosity over entropy ratio, below the bound from the anti-de Sitter conformal field theory conjecture. Some caveats on this result are discussed.