  We describe the anomalous phase transition of the emergence of the giant
connected component in scale-free networks growing under mechanism of
preferential linking. We obtain exact results for the size of the giant
connected component and the distribution of vertices among connected
components. We show that all the derivatives of the giant connected component
size $S$ over the rate $b$ of the emergence of new edges are zero at the
percolation threshold $b_c$, and $S \propto \exp\{-d(\gamma)(b-b_c)^{-1/2}\}$,
where the coefficient $d$ is a function of the degree distribution exponent
$\gamma$. In the entire phase without the giant component, these networks are
in a ``critical state'': the probability ${\cal P}(k)$ that a vertex belongs to
a connected component of a size $k$ is of a power-law form. At the phase
transition point, ${\cal P}(k) \sim 1/(k\ln k)^2$. In the phase with the giant
component, ${\cal P}(k)$ has an exponential cutoff at $k_c \propto 1/S$. In the
simplest particular case, we present exact results for growing exponential
networks.
