It was recently shown that estimating the Shannon entropy $H(p)$ of a discrete $k$-symbol distribution $p$ requires $\Theta(k/\log k)$ samples, a number that grows near-linearly in the support size. In many applications $H(p)$ can be replaced by the more general R\'enyi entropy of order $\alpha$, $H_\alpha(p)$. We determine the number of samples needed to es