Consider the problem of estimating the Shannon  entropy of a distribution on $ k$  elements from $ n$  independent  samples. We show that the minimax mean-square error is within  universal multiplicative constant factors of  $\left( \frac{n}{k \log n} \right)^{2} + \frac{\log^2 k}{n}$. This  implies the recent result of Valiant-Valiant [ 1 ] that the minimal  sample size for consistent entropy estimation scales according to $\Theta( \frac{k}{\log k} )$.  The apparatus of best polynomial approximation plays a key role in both the minimax lower bound and the construction of optimal estimators.