The room-temperature growth of Cu on the polar (0001)-Zn and (0001)-O surfaces of zinc oxide has been studied with scanning tunneling microscopy (STM). Copper grows on the (0001)-Zn surface as three-dimensional clusters even at low coverages (0.05-0.25 monolayers (ML)); two-dimensional (2D) islands are only observed at very low coverages (0.001-0.05 ML). The average size of the 3D clusters increases with coverage, and their density increases slowly. Surface roughness and sputter damage change the growth mode to more 2D-like. The Cu clusters are well-separated and exhibit a well-defined hexagonal shape. Equilibrium crystal shape analysis yields an apparent work of adhesion of 3.4 +/- 0.1 J/m(2) for the largest clusters. On the (0001)-O surface, formation of two-dimensional Cu clusters was observed at coverages of less than 0.1 ML.