  We consider the asymptotic behavior of solutions to the Cauchy problem for
the defocusing nonlinear Klein-Gordon equation (NLKG) with exponential
nonlinearity in the one spatial dimension with data in the energy space
$H^1(\mathbb{R}) \times L^2(\mathbb{R})$. We prove that any energy solution has
a global bound of the $L^6_{t,x}$ space-time norm, and hence scatters in
$H^1(\mathbb{R}) \times L^2(\mathbb{R})$ as $t\rightarrow\pm \infty$. The proof
is based on the argument by Killip-Stovall-Visan (Trans. Amer. Math. Soc. 364
(2012), no. 3, 1571--1631). However, since well-posedness in
$H^{1/2}(\mathbb{R}) \times H^{-1/2}(\mathbb{R})$ for NLKG with the exponential
nonlinearity holds only for small initial data, we use the $L_t^6
W^{s-1/2,6}_x$-norm for some $s>\frac{1}{2}$ instead of the $L_{t,x}^6$-norm,
where $W_x^{s,p}$ denotes the $s$-th order $L^p$-based Sobolev space.
