  Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank 2, and set
$\lambda=\Lambda_{1} - \Lambda_{2}$, where $\Lambda_{1}$, $\Lambda_{2}$ are the
fundamental weights. Denote by $V(\lambda)$ the extremal weight module of
extremal weight $\lambda$ with $v_\lambda$ the extremal weight vector, and by
$\mathcal{B}(\lambda)$ the crystal basis of $V(\lambda)$ with $u_\lambda$ the
element corresponding to $v_\lambda$. We prove that (i) $\mathcal{B}(\lambda)$
is connected, (ii) the subset $\mathcal{B}(\lambda)_{\mu}$ of elements of
weight $\mu$ in $\mathcal{B}(\lambda)$ is a finite set for every integral
weight $\mu$, and $\mathcal{B}(\lambda)_{\lambda} = \{u_\lambda\}$, (iii) every
extremal element in $\mathcal{B}(\lambda)$ is contained in the Weyl group orbit
of $u_\lambda$, (iv) $V(\lambda)$ is irreducible. Finally, we prove that the
crystal basis $\mathcal{B}(\lambda)$ is isomorphic, as a crystal, to the
crystal $\mathbb{B}(\lambda)$ of Lakshmibai-Seshadri paths of shape $\lambda$.
