Context. Water is the major component of the interstellar ice mantle. In interstellar ice, chemical reactivity is limited by the diffusion of the reacting molecules, which are usually present at abundances of a few percent with respect to water. Aims. We want to study the thermal diffusion of H2CO, NH3, HNCO, and CO in amorphous water ice experimentally to account for the mobility of these molecules in the interstellar grain ice mantle. Methods. In laboratory experiments performed at fixed temperatures, the diffusion of molecules in ice analogues was monitored by Fourier transform infrared spectroscopy. Diffusion coefficients were extracted from isothermal experiments using Fick's second law of diffusion. Results. We measured the surface diffusion coefficients and their dependence with the temperature in porous amorphous ice for HNCO, H2CO, NH3, and CO. They range from 10(-15) to 10(-11) cm(2) s(-1) for HNCO, H2CO, and NH3 between 110 K and 140 K, and between 5-8 x 10(-13) cm(2) s(-1) for CO between 35 K and 40 K. The bulk diffusion coefficients in compact amorphous ice are too low to be measured by our technique and a 10(-15) cm(2) s(-1) upper limit can be estimated. The amorphous ice framework reorganization at low temperature is also put in evidence. Conclusions. Surface diffusion of molecular species in amorphous ice can be experimentally measured, while their bulk diffusion may be slower than the ice mantle desorption kinetics.