This paper describes attempts to replace the present definition of the SI unit mass, the kilogram, by a new one, based on the atomic mass unit; the kilogram then becomes the mass of a certain number of silicon atoms. The related research is planned and is to be performed within the scope of a worldwide collaboration coordinated by the Working Group on the Avogadro Constant of the CIPM Consultative Committee for Mass and Related Quantities. This requires determination of the Avogadro constant, N_A, with a relative uncertainty close to 1 × 10⁻⁸. At present, the most important limiting factor is the uncertainty arising from observed differences between some primary density standards as well as the measurement of the molar mass of silicon, e.g. the determination of the natural isotopic composition. Improvements in crystal characterization and density determination are described that allow us to reach a relative uncertainty of about 10⁻⁷. A further reduction in the uncertainty is envisaged by measurements performed on a 99.99% enriched ²⁸Si single crystal.