We report first-principles calculations of acoustic surface plasmons on the (0001) surface of Be, as obtained in the random-phase approximation of many-body theory. The energy dispersion of these collective excitations has been obtained along two symmetry directions. Our results show a considerable anisotropy of acoustic surface plasmons, and underline the capability of experimental measurements of these plasmons to {it map} the electron-hole excitation spectrum of the quasi two-dimensional Shockley surface state band that is present on the Be(0001) surface.