  Let $\rho$ be a maximal representation of a uniform lattice
$\Gamma\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian
type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there
exists a holomorphic or antiholomorphic $\rho$-equivariant map from complex
hyperbolic space to the symmetric space associated to ${\rm SU}(p,q)$. This map
is moreover a totally geodesic homothetic embedding. In particular, up to a
representation in a compact subgroup of ${\rm SU}(p,q)$, the representation
$\rho$ extends to a representation of ${\rm SU}(n,1)$ in ${\rm SU}(p,q)$.
