  We show that the intersection exponents for planar Brownian motions are
analytic. More precisely, let $B$ and $B'$ be independent planar Brownian
motions started from distinct points, and define the exponent $\xi (1,
\lambda)$ by $$ E[P[B[0,t] \cap B'[0,t] = \emptyset | B[0,t]]^\lambda] \approx
t^{-\xi(1, \lambda)/2}, t \to \infty. $$ Then the mapping $\lambda \mapsto \xi
(1, \lambda)$ is real analytic in $(0,\infty)$. The same result is proved for
the exponents $\xi (k, \lambda)$ where $k$ is a positive integer. In
combination with the determination of $\xi (k, \lambda)$ for integer $k \ge 1$
and real $\lambda \ge 1$ in our previous papers, this gives the value of $\xi
(k, \lambda)$ also for $\lambda \in (0,1)$ and the disconnection exponents
$\lim_{\lambda \searrow 0} \xi (k, \lambda)$. In particular, it shows that
$\lim_{\lambda \searrow 0} \xi(2, \lambda) = 2/3$ and concludes the proof of
the following result that had been conjectured by Mandelbrot: the Hausdorff
dimension of the outer boundary of $B[0,1]$ is 4/3 almost surely.
