We report the first precise measurement of a β-recoil correlation from a radioactive noble gas (⁶He) confined via a magneto-optical trap. The measurement is motivated by the search for exotic tensor-type contributions to the charged weak current. Interpreted as tensor currents with right-handed neutrinos, the measurements yield |C_T/C_A|² ≤ 0.022 (90% confidence limit, C.L.). On the other hand, for left-handed neutrinos the limits are 0.007 < C_T/C_A < 0.111 (90% C.L.).  The sensitivity of the present measurement is mainly limited by experimental uncertainties in determining the time response properties and the distance between the atom cloud and the microchannel plate used for recoil ion detection.