Opacity is a property of many plasmas, and it is normally expected that if an emission line in a plasma becomes optically thick, its intensity ratio to that of another transition that remains optically thin should decrease. However, radiative transfer calculations undertaken both by ourselves and others predict that under certain conditions the intensity ratio of an optically thick to thin line can show an increase over the optically thin value, indicating an enhancement in the former. These conditions include the geometry of the emitting plasma and its orientation to the observer. A similar effect can take place between lines of differing optical depth. Previous observational studies have focused on stellar point sources, and here we investigate the spatially-resolved solar atmosphere using measurements of the I(1032 A)/I(1038 A) intensity ratio of O VI in several regions obtained with the Solar Ultraviolet Measurements of Emitted Radiation (SUMER) instrument on board the Solar and Heliospheric Observatory (SoHO) satellite. We find several I(1032 A)/I(1038 A) ratios observed on the disk to be significantly larger than the optically thin value of 2.0, providing the first detection (to our knowledge) of intensity enhancement in the ratio arising from opacity effects in the solar atmosphere. Agreement between observation and theory is excellent, and confirms that the O VI emission originates from a slab-like geometry in the solar atmosphere, rather than from cylindrical structures.