Let ${\rm Rat}_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of degree $d\geq 2$ , and let $\mu_f$ be the measure of maximal entropy for $f\in{\rm Rat}_d$ . The map of measures $f\mapsto\mu_f$ is known to be continuous on ${\rm Rat}_d$ , and it is shown here to extend continuously to the boundary of ${\rm Rat}_d$ in $\overline{\rm Rat}_d \simeq {\bf P}{\rm H}^0({\bf P}^1\times {\bf P}^1, {\cal O}(d,1))\simeq {\bf P}^{2d+1}$ , except along a locus $I(d)$ of codimension $d+1$ . The set $I(d)$ is also the indeterminacy locus of the iterate map $f\mapsto f^n$ for every $n\geq 2$ . The limiting measures are given explicitly, away from $I(d)$ . The degenerations of rational maps are also described in terms of metrics of nonnegative curvature on the Riemann sphere; the limits are polyhedral