We have searched for antihelium nuclei in cosmic rays using the data obtained from balloon flights of the BESS magnetic spectrometer. The search was mainly based on track-quality selection, followed by rigidity analysis, and on the time-of-flight and dE/dx measurements by the scintillation counter hodoscope. We analysed all the data collected during 1993-2000 with a common analysis procedure. No antihelium nuclei events were found in the energy range from 1 to 14 GV. In order to determine a new upper limit, we have simulated the loss in the air and in the instrument of He (He) using the GEANT/GHEISHA code. Combined with the data collected in 1993 through 2000, a new 95 % confidence upper limit for the ratio of He/He at the top of the atmosphere of 6.8 × 10-7 has been obtained to be after correcting for the interactions in the air and in the instruments.