With the axiom of choice, it is easy to show that there is an uncountable proper subgroup of the additive group of real numbers. Namely, if $H$ is a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$, then for any $x \in H$, the $\mathbb{Q}$-vector subspace of $\mathbb{R}$ generated (or spanned) by $H \setminus \{x\}$ is an uncountable proper subgroup.
But, could an uncountable proper subgroup of the additive group of real numbers still be proven to exist without the axiom of choice?