  We report the first measurement of target single-spin asymmetries (A$_N$) in
the inclusive hadron production reaction,
$e~$+$~^3\text{He}^{\uparrow}\rightarrow h+X$, using a transversely polarized
$^3$He target. The experiment was conducted at Jefferson Lab in Hall A using a
5.9-GeV electron beam. Three types of hadrons ($\pi^{\pm}$, $\text{K}^{\pm}$
and proton) were detected in the transverse hadron momentum range 0.54 $<p_T<$
0.74 GeV/c. The range of $x_F$ for pions was -0.29 $<x_F<$ -0.23 and for kaons
-0.25 $<x_F<$-0.18. The observed asymmetry strongly depends on the type of
hadron. A positive asymmetry is observed for $\pi^+$ and $\text{K}^+$. A
negative asymmetry is observed for $\pi^{-}$. The magnitudes of the asymmetries
follow $|A^{\pi^-}| < |A^{\pi^+}| < |A^{K^+}|$. The K$^{-}$ and proton
asymmetries are consistent with zero within the experimental uncertainties. The
$\pi^{+}$ and $\pi^{-}$ asymmetries measured for the $^3$He target and
extracted for neutrons are opposite in sign with a small increase observed as a
function of $p_T$.
