We report a 0.08 % measurement of the bound neutron scattering length of $^4$He using neutron interferometry. The result is $b = (3.0982 pm 0.0021 mbox{ [stat]} pm 0.0014 mbox{ [sys]}) mbox{ fm}$. The corresponding free atomic scattering length is $a = (2.4746 pm 0.0017 mbox{ [stat]} pm 0.0011 mbox{ [sys]}) mbox{ fm}$. With this result the world average becomes $b = (3.0993 pm 0.0025)$ fm, a 2 % downward shift and a reduction in uncertainty by more than a factor of six. Our result is in disagreement with a previous neutron interferometric measurement but is in good agreement with earlier measurements using neutron transmission.
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