High power, relativistic electron beams from energy recovery linacs have great potential to realize new experimental paradigms for pioneering innovation in fundamental and applied research. A major design consideration for this new generation of experimental capabilities is the understanding of the halo associated with these bright, intense beams. In this Letter, we report on measurements performed using the 100 MeV, 430 kWatt CW electron beam from the energy recovery linac at the Jefferson Laboratory's Free Electron Laser facility as it traversed a set of small apertures in a 127 mm long aluminum block. Thermal measurements of the block together with neutron measurements near the beam-target interaction point yielded a consistent understanding of the beam losses. These were determined to be 3 ppm through a 2 mm diameter aperture and were maintained during a 7 hour continuous run.