  We construct a natural extension for each of Nakada's $\alpha$-continued
fractions and show the continuity as a function of $\alpha$ of both the entropy
and the measure of the natural extension domain with respect to the density
function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le
1$, the product of the entropy with the measure of the domain equals $\pi^2/6$.
As a key step, we give the explicit relationship between the $\alpha$-expansion
of $\alpha-1$ and of $\alpha$.
