  We investigate the ground state of the $J_1$-$J_2$ spin-1/2 chain with
$J_1<0$ and $J_2>0$ in the case that the nearest-neighbor $J_1$ interaction in
the $z$-direction has a weak anisotropy as $J_1 (1-\alpha)$. We perform a
perturbational analysis for small $\alpha$ and $\lambda \equiv J_2- |J_1|/4$
with the exact solution of the unperturbed ground state for $\alpha = \lambda =
0$. The scaling property of the ground state energy is examined in detail. By
the numerical diagonalization analysis of finite size systems, we found the
phase boundary equation between the spin fluid and dimer phases as $\alpha_{\rm
c} = 14 \lambda_{\rm c}^{4/3}$.
