We present the first measurement of the α-β-ν angular correlation in the Gamow-Teller β^{+} decay of ^{8}B. This was accomplished using the Beta-decay Paul Trap, expanding on our previous work on the β^{-} decay of ^{8}Li. The ^{8}B result is consistent with the V-A electroweak interaction of the standard model and, on its own, provides a limit on the exotic right-handed tensor current relative to the axial-vector current of |C_{T}/C_{A}|^{2}<0.013 at the 95.5% confidence level. This represents the first high-precision angular correlation measurements in mirror decays and was made possible through the use of an ion trap. By combining this ^{8}B result with our previous ^{8}Li results, we demonstrate a new pathway for increased precision in searches for exotic currents.