A computational algorithm is described for direct numerical simulation (DNS) of turbulent mixing of two incompressible miscible fluids having greatly differing densities. The algorithm uses Fourier pseudo-spectral methods to compute spatial derivatives and a fractional step method involving the third-order Adams-Bashforth-Moulton predictor-corrector scheme to advance the solution in time. The pressure projection technique is shown to eliminate stability problems, previously observed, when the ratio of the densities in the two streams is as high as 35. The algorithm is investigated in detail for mixing in isotropic homogeneous turbulence of two fluids with a density ratio of 10. The limit on the density ratio is imposed so that the flow is both everywhere turbulent and spatially resolved. Both fluids have the same molecular viscosities, the nominal Schmidt number is 0.7, and the initial nominal Reynolds number based on the integral length scale and the rms velocity is 158. No body force is considered. It is shown that the pressure projection scheme does not limit the temporal accuracy of the solution when periodic boundary conditions are used, but that it significantly affects the stability of the simulations. It is also shown that the rate at which turbulence kinetic energy dissipates averaged for the whole computational domain is almost unaffected by density ratio.