  Let $P(n,M)$ be a graph chosen uniformly at random from the family of all
labeled planar graphs with $n$ vertices and $M$ edges. In the paper we study
the component structure of $P(n,M)$. Combining counting arguments with analytic
techniques, we show that there are two critical periods in the evolution of
$P(n,M)$. The first one, of width $\Theta(n^{2/3})$, is analogous to the phase
transition observed in the standard random graph models and takes place for
$M=n/2+O(n^{2/3})$, when the largest complex component is formed. Then, for
$M=n+O(n^{3/5})$, when the complex components cover nearly all vertices, the
second critical period of width $n^{3/5}$ occurs. Starting from that moment
increasing of $M$ mostly affects the density of the complex components, not its
size.
