Carboranethiol molecules self-assemble into upright molecular monolayers on Au{111} with aligned dipoles in two dimensions. The positions and offsets of each molecule's geometric apex and local dipole moment are measured and correlated with sub-Ångström precision. Juxtaposing simultaneously acquired images, we observe monodirectional offsets between the molecular apexes and dipole extrema. We determine dipole orientations using efficient new image analysis techniques and find aligned dipoles to be highly defect tolerant, crossing molecular domain boundaries and substrate step edges. The alignment observed, consistent with Monte Carlo simulations, forms through favorable intermolecular dipoleÀdipole interactions.