Let $\gamma:\mathbb R \to G$ denote the one parameter sub-group, and $H$ its image in $G$.
The closure $\overline H$ of $H$ is an abelian (connected) Lie group, and so can be written (non-canonically) in the form $\overline H = V \times T$, where $V$ is a vector-space and $T$ a torus. We want to show, if $H\not=\overline H$, that $V= 0$. Suppose $p=(v,t)\in\overline H \setminus H$. The projection of $\gamma$ on $V$ is of the form $x \mapsto xw$, for some non-zero $w\in V$ (non-zero, for otherwise $H$ is not dense in $\overline H$ (!), unless $V=0$), so there is an open neighborhood $B$ of $p$, with compact closure, and a bound $R$ such that $|x|>R$ implies $\gamma (x) \not \in \overline B$. The set $\gamma([-R,R])$ is compact, and contains $p$ in its closure, so contains $p$. Therefore $p\in H$. Contradiction! Therefore $V=0$.