On 1987 August 22 a balloon flight was conducted using the Goddard Space Flight Center Low-Energy Antiproton configuration of the New Mexico State University balloon-borne magnet spectrometer. The launch site was Prince Albert, Saskatchewan, Canada. The balloon flew at an average atmospheric depth of 4.7 g cm-2 for more than 22 hr. During this period a sample of 4.2 × 104 helium nuclei was gathered. No antihelium candidates were found in this sample. The resultant upper limit for the ratio of antihelium to helium in cosmic rays over the rigidity interval from 1 to 25 GV/c is 9 × 10-5 at 95% confidence.This limit is below the predicted level, assuming equal matter and antimatter in the extragalactic cosmic rays.