We have searched for antihelium in the galactic cosmic rays using data obtained on the flight of the Balloon-borne Experiment with a Superconducting Spectrometer (BESS) launched from Lynn Lake, Manitoba, in northern Canada on 1995 July 25. The balloon reached an altitude of 36.5 km with a residual overburden of 5 g cm-2. The total observation time was 12.2 hr with a live-time fraction of 0.61. No antihelium was observed after selections based on dE/dx and event quality cuts at rigidities between 1.6 and 16 GV/c; we infer that 536,420 helium nuclei survived the same cuts. This result leads to an upper limit to the He/He abundance ratio of 8.1 × 10-6 (95% confidence level), a factor of 2.7 over the lowest previous limit. Because this limit is for the first time at the level where predicted limits become astrophysically interesting (Ahlen et al.), we have reexamined the transport of cosmic rays in intergalactic space. We show that little can be learned about distant (>10 Mpc) domains of antimatter from a null result.