  We investigate an $S=1/2$ two-leg spin ladder with a cyclic four-spin
exchange interaction whose interaction constant is denoted by $J_4$, by using
the density matrix renormalization group method. The interchain and the
intrachain interaction constant are denoted by $J_{rung}$ and $J_{\rm leg}$,
respectively and assumed to be antiferromagnetic. It turns out that a spin gap
between the singlet ($S_{tot}^z=0$) and the triplet ($S_{tot}^z=1$) states
vanishes at $J_4/J_{leg} \simeq 0.3$ for $J_{rung}=J_{leg}$. This result is in
contrast with the fact that the $S=1/2$ antiferromagnetic Heisenberg ladder,
that is the case of $J_{rung} \neq J_{leg}, J_4=0$, has a spin gap for all
nonzero value of interchain interaction $J_{\rm rung}>0$. We find a larger
value of the correlation length for the spin-pair correlation function than a
linear size $L$ of the system at $J_4/J_{\rm leg}=0.3$ and $J_{rung}=J_{leg}$:
the correlation length $\xi$ is about 204 times of the lattice constant for
L=84 for these values of interactions. We also find that the string correlation
function decays rather algebraically than exponentially at $J_4/J_{leg}=0.3$
and $J_{rung}=J_{leg}$. These results suggest that there is a quantum phase
transition at $J_4/J_{leg} \simeq 0.3$ for $J_{rung}=J_{leg}$. We estimate a
phase boundary where the spin gap vanishes in a $J_4/J_{leg}-J_{rung}/J_{leg}$
plane and obtain a consistent result with that by a perturbation theory for
$J_{\rm rung}/J_{leg}>1$.
