Let $R\subseteq\mathbb{Q}$ be a local subring with maximal ideal $\mathfrak{m}$. Is $R$ a valuation ring of $\mathbb Q$?
$R$ is a valuation ring iff its ideals are linearly ordered. But I'm stuck here. Is the fact that every subring of rationals is in 1-1 correspondence to a set of primes useful here?
$\mathbb{Z}\subset R\subset \mathbb{Q}$. Look at $\mathfrak{m}\cap \mathbb{Z}$. This is a prime ideal, so it is either $0$ or $p\mathbb{Z}$ for a prime $p$. Check (easily) that in the first case $R=\mathbb{Q}$ and in the second case $R$ is the localization of $\mathbb{Z}$ at $p\mathbb{Z}$ and thus a dvr.