The atomic structure of quasicrystals(1)-solids with long-range order, but non-periodic atomic lattice structure-is often described as the three-dimensional generalization of the planar two-tile Penrose pattern(2). Recently, an alternative model has been proposed(3-5) that describes such structures in terms of a single repeating unit(3-5)-the three-dimensional generalization of a pattern composed of identical decagons. This model is similar in concept to the unit-cell description of periodic crystals, with the decagon playing the role of a ＇quasi-unit cell＇. But, unlike the unit cells in periodic crystals, these quasi-unit cells,overlap their neighbours, in the sense that they share atoms. Nevertheless, the basic concept of unit cells in both periodic crystals and quasicrystals is essentially the same : solving the entire atomic structure of the solid reduces to determining the distribution of atoms in the unit cell. Here we report experimental evidence for the quasi-unit-cell model by solving the structure of the decagonal quasicrystal Al72Ni20Co8. The resulting structure is consistent with images obtained by electron and X-ray diffraction, and agrees with the measured stoichiometry, density and symmetry of the compound. The quasi-unit-cell model provides a significantly better fit to these results than all previous alternative models, including Penrose tiling.