The orientation of molecules is crucial in many chemical processes. Here, we report how single dipolar molecules can be oriented with maximum precision using the electric field of a scanning tunneling microscope. Rotation is found to occur around a fixed pivot point that is caused by the specific interaction of an oxygen atom in the molecule with the Ag(111) surface. Both directions of rotation are realized at will with 100% directionality. Consequently, the internal dipole moment of an individual molecule can be spatially mapped via its behavior in an applied electric field. The importance of the oxygen-surface interaction is demonstrated by the addition of a silver atom between a single molecule and the surface and the consequent loss of the pivot point.