We present results of Brownian dynamics simulations of aqueous 1-1 electrolyte solutions in the 1-molar concentration range. The electrical conductivity and the self-diffusion coefficients obtained from the simulations are compared to experimental data. The interaction potential between the ions is modeled by pairwise repulsive 1/r(n) soft-core interactions (n = 9 or n = 12) and Coulomb forces. We take into account hydrodynamic interactions and integrate the stochastic equations of motion with large time steps of about 100 femtoseconds, combined with an acceptance criterion known from the Smart Monte Carlo method. In this way, details of the dynamics of particles in close contact are not considered and the short-ranged repulsive forces act effectively as constraint forces preventing overlap configurations. The lengths of the performed simulations (about 10 nanoseconds) and the number of ions (216) allow to obtain single particle as well as collective transport coefficients with sufficient precision. For this purpose we use Kubo expressions which can be applied on the mesoscopic time scale of Brownian dynamics simulations. It is shown that hydrodynamic interactions must be taken into account to obtain agreement with the experimental data. They lower the electrical conductivity, as expected, but increase the self-diffusion coefficients, confirming a recent finding for colloids.