  We present an analytical solution of the Ginzburg's $\Psi$-theory for the
behavior of the Casimir force in a film of $^4$He in equilibrium with its vapor
near the superfluid transition point, and we revisit the corresponding
experiments in light of our findings. We find reasonably good agreement between
the $\Psi$-theory predictions and the experimental data. Our calculated force
is attractive, and the largest absolute value of the scaling function is
$1.848$, while experiment yields $1.30$. The position of the extremum is
predicted to be at $x=(L/\xi_0)(T/T_\lambda-1)^{1/\nu}=\pi$, while experiment
is consistent with $x=3.8$. Here $L$ is the thickness of the film, $T_\lambda$
is the bulk critical temperature and $\xi_0$ is the correlation length
amplitude of the system for $T>T_\lambda$.

Statistical Mechanics