Using TRIUMFs neutral atom trap, TRINAT, for nuclear $beta$ decay, we have measured the $beta$ asymmetry with respect to the initial nuclear spin in $^{37}mathrm{K}$ to be $A_beta=-0.5707(13)_mathrm{syst}(13)_mathrm{stat}(5)_mathrm{pol}$, a 0.3% measurement. This is the best relative accuracy of any $beta$-asymmetry measurement in a nucleus or the neutron, and is in agreement with the standard model prediction $-0.5706(7)$. We compare constraints on physics beyond the standard model with other $beta$-decay measurements, and improve the value of $V_mathrm{ud}$ measured in this mirror nucleus by a factor of 4.
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