  We investigate theoretically production cross sections of the $J^\pi=$ 0$^+$
ground state of a $^4_\Lambda$He hypernucleus in the $^4$He($\pi$,~$K$)
reaction with a distorted-wave impulse approximation using the optimal
Fermi-averaged $\pi N\to K\Lambda$ $t$ matrix. We demonstrate the sensitivity
of the production cross sections to the $\Lambda$ wavefunctions obtained from
$3N$-$\Lambda$ potentials and to meson distorted waves in eikonal distortions.
It is shown that the calculated laboratory cross sections of the 0$^+$ ground
state in $^4_\Lambda$He amount to $d\sigma/d\Omega_{\rm lab} \simeq$ 11
$\mu$b/sr at $p_{\pi}=$ 1.05 GeV/$c$ in the $K$ forward direction because of an
advantage of the use of the s-shell target nucleus such as $^4$He. The
importance of the recoil effects and the energy dependence of the $\pi N\to
K\Lambda$ cross sections is also discussed.
