In high intensity linacs emittance exchange driven by space charge coupling may lead to the well-known "equipartitioning" phenomenon if the stop band at sigma(parallel) = sigma(perpendicular) is crossed at sufficiently slow rate. This Letter is the first experimental evidence of this phenomenon in a high intensity linear accelerator, here the UNILAC at GSI. Measurements of emittances at the entrance and exit of one drift tube linac tank comprising 15 lattice cells are taken for a set of transverse and longitudinal tunes. The onset of exchange on the stop band of previously derived "stability charts" confirms theoretical predictions. The measured transverse emittance growth also compares well with results from the beam dynamics simulation codes DYNAMION and TRACEWIN.