A precision measurement of the β^{+} decay of ^{8}B was performed using the Beta-decay Paul Trap to determine the β-ν angular correlation coefficient a_{βν}. The experimental results were combined with new ab initio symmetry-adapted no-core shell-model calculations to yield the second-most precise measurement from Gamow-Teller decays, a_{βν}=-0.3345±0.0019_{stat}±0.0021_{syst}. This value agrees with the standard model value of -1/3 and improves uncertainties in ^{8}B by nearly a factor of 2. By combining results from ^{8}B and ^{8}Li, a tight limit on tensor current coupling to right-handed neutrinos was obtained. A recent global evaluation of all other precision β decay studies suggested a nonzero value for right-handed neutrino coupling in contradiction with the standard model at just above 3σ. The present results are of comparable sensitivity and do not support this finding.