In an extensive computer simulation study, the transport coefficients of the Lennard-Jones model fluid were determined with high accuracy from equilibrium molecular-dynamics simulations. In the frame of time-correlation function theory, the generalized Einstein relations were employed to evaluate the transport coefficients. This second of a series of four papers presents the results for the self-diffusion coefficient, and discusses and interprets the behavior of this transport coefficient in the fluid region of the phase diagram. The uncertainty of the self-diffusion data is estimated to be 1% in the gas region and 0.5% at high-density liquid states. With the very accurate data, even fine details in the shape of the self-diffusion isotherms are resolved, and the previously little-investigated behavior of the self-diffusion coefficient at low-density gaseous states is analyzed in detail. Finally, aspects of the mass transport mechanisms on the molecular scale are explored by an analysis of the velocity autocorrelation functions.