Radioactive decay of uranium and thorium produces 4He, whereas 3He in the Earth's mantle is not produced by radioactive decay and was only incorporated during accretion-that is, it is primordial. 3He/4He ratios in many ocean-island basalts (OIBs) that erupt at hotspot volcanoes, such as Hawaii and Iceland, can be up to sixfold higher than in mid-ocean ridge basalts (MORBs). This is inferred to be the result of outgassing by melt production at mid-ocean ridges in conjunction with radiogenic ingrowth of 4He, which has led to a volatile-depleted upper mantle (MORB source) with low 3He concentrations and low 3He/4He ratios. Consequently, high 3He/4He ratios in OIBs are conventionally viewed as evidence for an undegassed, primitive mantle source, which is sampled by hot, buoyantly upwelling deep-mantle plumes. However, this conventional model provides no viable explanation of why helium concentrations and elemental ratios of He/Ne and He/Ar in OIBs are an order of magnitude lower than in MORBs. This has been described as the 'helium concentration paradox' and has contributed to a long-standing controversy about the structure and dynamics of the Earth's mantle. Here we show that the helium concentration paradox, as well as the full range of noble-gas concentrations observed in MORB and OIB glasses, can self-consistently be explained by disequilibrium open-system degassing of the erupting magma. We show that a higher CO2 content in OIBs than in MORBs leads to more extensive degassing of helium in OIB magmas and that noble gases in OIB lavas can be derived from a largely undegassed primitive mantle source.