  In this paper, we give a new class of rigid Coxeter groups. Let $(W,S)$ be a
Coxeter system. Suppose that (0) for each $s,t\in S$ such that $m(s,t)$ is
even, $m(s,t)=2$, (1) for each $s\neq t\in S$ such that $m(s,t)$ is odd,
$\{s,t\}$ is a maximal spherical subset of $S$, (2) there does not exist a
three-points subset $\{s,t,u\}\subset S$ such that $m(s,t)$ and $m(t,u)$ are
odd, and (3) for each $s\neq t\in S$ such that $m(s,t)$ is odd, the number of
maximal spherical subsets of $S$ intersecting with $\{s,t\}$ is at most two,
where $m(s,t)$ is the order of $st$ in the Coxeter group $W$. Then we show that
the Coxeter group $W$ is rigid. This is an extension of a result of
D.Radcliffe.
