In two long-duration balloon flights over Antarctica, the Balloon-borne Experiment with a Superconducting Spectrometer (BESS) collaboration has searched for antihelium in the cosmic radiation with the highest sensitivity reported. BESS-Polar I flew in 2004, observing for 8.5 days. BESS-Polar II flew in 2007-2008, observing for 24.5 days. No antihelium candidate was found in BESS-Polar I data among 8.4×10(6) |Z|=2 nuclei from 1.0 to 20 GV or in BESS-Polar II data among 4.0×10(7) |Z|=2 nuclei from 1.0 to 14 GV. Assuming antihelium to have the same spectral shape as helium, a 95% confidence upper limit to the possible abundance of antihelium relative to helium of 6.9×10(-8)} was determined combining all BESS data, including the two BESS-Polar flights. With no assumed antihelium spectrum and a weighted average of the lowest antihelium efficiencies for each flight, an upper limit of 1.0×10(-7) from 1.6 to 14 GV was determined for the combined BESS-Polar data. Under both antihelium spectral assumptions, these are the lowest limits obtained to date.