  Given a non-increasing function $\psi\colon\mathbb{N}\to\mathbb{R}^+$ such
that $s^{\frac{n+1}{n}}\psi(s)$ tends to zero as $s$ goes to infinity, we show
that the set of points in $\mathbb{R}^n$ that are exactly $\psi$-approximable
is non-empty, and we compute its Hausdorff dimension. For $n\geq 2$, this
answers questions of Jarn\'{i}k and of Beresnevich, Dickinson, and Velani.
