The diffusion of molecules in interstellar ice is a fundamental phenomenon to take into account while studying the formation of complex molecules in this ice. This work presents a theoretical study on the diffusion of H2O, NH3, CO2, CO, and H2CO in the bulk of a low density amorphous (LDA) ice, while taking into account the physical conditions prevailing in space, i.e. temperatures below 150 K and extremely low pressure. This study was undertaken by means of molecular dynamics simulations. For CO2 for which no experimental data were available we conducted our own experiments. From our calculations we show that, at low temperatures, the diffusion of molecules in the bulk of a LDA ice is driven by the self-diffusion of water molecules in the ice. With this study we demonstrate that molecular dynamics allows the calculation of diffusion coefficients for small molecules in LDA ice that are convincingly comparable to experimentally measured diffusion coefficients. We also provide diffusion coefficients for a series of molecules of astrochemical interest.