Self-avoiding random surfaces on a cubic lattice are studied by extensive Monte Carlo sampling. The surfaces have an empty boundary and the topology of a 2-sphere. An oct-tree data-structure allows good statistics to be obtained for surfaces whose plaquette number is up to an order of magnitude greater than in previous investigations. The new simulation strategy is explained in detail and compared with previous ones. The critical plaquette fugacity, mu ^-1, and the entropic exponent, theta , are determined by maximum likelihood methods and by logarithmic plots of the average surface area versus fugacity. The latter approach, which produces results having much better convergence by taking advantage of the scaling properties of several runs at various fugacities, leads to the estimates mu =1.729+or-0.036 and theta =1.500+or-0.026. Linear regression estimates for the radius of gyration exponent give v=0.509+or-0.004, while the asymptotic ratio of surface area over average volume enclosed approaches a finite value 3.18+or-0.03. The results give strong corroborating evidence that this long-controversial problem belongs to the universality class of branched polymers.