We analyze helium (He) and neon (Ne) isotope data sets from the southeast Pacific sector of the Southern Ocean collected in 1992 and 1994 and describe a new method to estimate glacial meltwater fluxes independent of previous approaches. The waters sampled during these cruises reveal large atmospheric He and Ne excess concentrations in the upper 500 m that are derived predominantly from the dissolution of air contained in glacial ice during melting underneath and at the front of floating ice shelves at elevated hydrostatic pressure. On the stations sampled, the meltwater fractions are largest near the northern and southern ice shelf fronts of George VI Ice Shelf (1.5--2.5%), consistent with prior inferences from δ18O. The inventories of 4He and Ne with local glacial meltwater origin south of 70¿S between 210¿E and 295¿E are (2.5 ¿ 0.7)1011 cm3STP 4He and (7.9 ¿ 2.7)1011 cm3STP Ne, respectively. The associated meltwater volumes are 430 ¿ 120 Gt for 4He and 400 ¿ 140 Gt for Ne. For an estimated seawater residence time of 2 years on the continental shelf, the amounts of glacial meltwater translate into bulk melting rates of 215 ¿ 170 Gt yr-1 for 4He and 200 ¿ 170 Gt yr-1 for Ne. The values derived from 4He are expected to be larger than those obtained from Ne because of the terrigenic He component, which accounts for 10--15% of the He excess in this region. Our regional melting rate of 200--215 Gt yr-1 is comparable to a prior estimate but subject to uncertainties resulting from the station distribution, the air-sea gas exchange, and the estimate of the net transport through the study region.