The three-dimensional momentum density of annihilating electron - positron pairs has been studied for a single Al - Li - Cu icosahedral quasicrystal. A direct Fourier transform method is employed to reconstruct the three-dimensional momentum density from measurements of the two-dimensional angular correlation of positron annihilation radiation (2D-ACAR). The crystallographic anisotropy in the momentum density is observed to be very small. The asphericity of the Fermi surface is not found explicitly within the experimental resolution in the momentum space. The features of the three-dimensional electron - positron momentum density agree with those obtained by means of Compton profile measurement. It is suggested that a strong lattice - electron interaction at the Fermi level occurs in this icosahedral phase.