First measurements of the breakdown threshold in a dielectric subjected to GV/m wakefields produced by short (30-330 fs), 28.5 GeV electron bunches have been made. Fused silica tubes of 100 microm inner diameter were exposed to a range of bunch lengths, allowing surface dielectric fields up to 27 GV/m to be generated. The onset of breakdown, detected through light emission from the tube ends, is observed to occur when the peak electric field at the dielectric surface reaches 13.8+/-0.7 GV/m. The correlation of structure damage to beam-induced breakdown is established using an array of postexposure inspection techniques.