On totally ordered abelian groups having exactly one convex subgroup

Question

Let $(G, <)$ be a totally ordered abelian group. Let us call a proper subgroup $H$ of $G$ to be convex if for every $a \in H$, $[a,-a]:=\{x\in G : -a \le x \le a\} \subseteq H$. If $G$ has exactly one convex subgroup (namely the trivial subgroup containing identity only), then does there exist an injective group homomorphism of $G$ into $(\mathbb R,+)$ ? If true, then can we find such an injective group homomorphism which is also order preserving , where $\mathbb R$ is equipped with the usual order ?