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We report the first precise measurement of a $\beta$-recoil correlation from a radioactive noble gas ($^6{\rm He}$) 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: $|C_T/C_A|^2\le 0.022$ (90\% 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 micro-channel plate used for recoil ion detection.