One can prove Popescu's theorem directly in this special case (due to the two strong assumptions -- characteristic $0$ and one-dimensionality -- present here). The basic point is that any 'singular' subalgebra $A \subset R$ may be resolved by a proper birational map $X \to \mathrm{Spec}(A)$ by Hironaka, and that the map $\mathrm{Spec}(R) \to \mathrm{Spec}(A)$ lifts to $X$ by the valuative criterion.
Write $R$ as a filtered colimit of finitely generated $\mathbb{Q}$-subalgebras $A_i \subset R$, so $\mathrm{Spec}(R) = \lim \mathrm{Spec}(A_i)$. Each $A_i$ is a domain, and the map $\mathrm{Spec}(R) \to \mathrm{Spec}(A_i)$ maps the generic point to the generic point. For each $i$, consider the cofiltered projective system $ C_i := \{X_{i,j} \to \mathrm{Spec}(A_i)\}$ of all proper maps which are isomorphisms over the generic point of the target. Pullback along $\mathrm{Spec}(A_{i'}) \to \mathrm{Spec}(A_i)$ gives a functor $C_i \to C_{i'}$, so we get a two-variable cofiltered projective system $\{X_{i,j}\}$ of schemes. Also, the map $\mathrm{Spec}(R) \to \mathrm{Spec}(A_i)$ admits a unique lift to $\mathrm{Spec}(R) \to X_{i,j}$ for each $j$ by the valuative criterion for properness. Let $B_{i,j}$ be the local ring of $X_{i,j}$ at the image of the closed point of $\mathrm{Spec}(R)$, so there is a natural map $B_{i,j} \to R$ extending the given map $A_i \to R$. Varying over the projective system, we get a filtered system $\{B_{i,j}\}$ of rings with a map to $R$. One checks that $\mathrm{colim} B_{i,j} \simeq R$; the surjectivity is completely trivial, while injectivity comes from the fact that $B_{i,j}$'s are domains birational to $A_i$ for a cofinal set of $j$'s. Moreover, by resolutions, for a fixed $i$, the $B_{i,j}$'s are smooth over $\mathbb{Q}$ for a cofinal set of $j$'s. Thus, we have expressed $R$ as a filtered colimit of smooth $\mathbb{Q}$-algebras.
