Domain strictly contained in the intersection of localizations at the primes of height one

Take $R=k[x^4,x^3y,xy^3,y^4]$.

There must be an easier example, but here is one I found after searching a bit, Ex. 7.2 in this arXiv paper by Epstein and Shapiro. It is sadly not finitely generated nor noetherian, but it is normal!

Let $k$ be a field and $R = k[x,y,y/x,y/x^2,y/x^3,\ldots] \subset k(x,y)$, which you might have seen as part of Karl Schwede's construction of a non-discrete valuation ring. The ideal $\mathfrak{m} = (x)$ is a height two principal prime ideal by this answer, hence $$ x^{-1} \in \left(\bigcap_{\operatorname{ht}\mathfrak{p} = 1} R_\mathfrak{p}\right) \setminus R. $$