  Using TRIUMF's 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.
