The first measurements of light antinucleus production in Au+Au collisions at the Relativistic Heavy-Ion Collider are reported. The observed production rates for d¯ and ³He¯ are much larger than in lower energy nucleus-nucleus collisions. A coalescence model analysis of the yields indicates that there is little or no increase in the antinucleon freeze-out volume compared to collisions at CERN SPS energy. These analyses also indicate that the ³He¯ freeze-out volume is smaller than the d¯ freeze-out volume.