There is now strong evidence that some stars have been born with He mass fractions as high as $Y approx 0.40$ (e.g., in $omega$ Centauri). However, the advanced evolution, chemical yields, and final fates of He-rich stars are largely unexplored. We investigate the consequences of He-enhancement on the evolution and nucleosynthesis of intermediate-mass asymptotic giant branch (AGB) models of 3, 4, 5, and 6 M$_odot$ with a metallicity of $Z = 0.0006$ ([Fe/H] $approx -1.4$). We compare models with He-enhanced compositions ($Y=0.30, 0.35, 0.40$) to those with primordial He ($Y=0.24$). We find that the minimum initial mass for C burning and super-AGB stars with CO(Ne) or ONe cores decreases from above our highest mass of 6 M$_odot$ to $sim$ 4-5 M$_odot$ with $Y=0.40$. We also model the production of trans-Fe elements via the slow neutron-capture process (s-process). He-enhancement substantially reduces the third dredge-up efficiency and the stellar yields of s-process elements (e.g., 90% less Ba for 6 M$_odot$, $Y=0.40$). An exception occurs for 3 M$_odot$, where the near-doubling in the number of thermal pulses with $Y=0.40$ leads to $sim$ 50% higher yields of Ba-peak elements and Pb if the $^{13}$C neutron source is included. However, the thinner intershell and increased temperatures at the base of the convective envelope with $Y=0.40$ probably inhibit the $^{13}$C neutron source at this mass. Future chemical evolution models with our yields might explain the evolution of s-process elements among He-rich stars in $omega$ Centauri.
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