  Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges.
Erd\H{o}s and R\'enyi (1960) conjectured that the limit
  $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists
and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994)
proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower
and upper bounds for this probability.
  In this paper we determine the exact probability of a random graph being
planar near the critical point $M=n/2$. For each $\lambda$, we find an exact
analytic expression for
  $$ p(\lambda) = \lim_{n \to \infty} \Pr{G(n,\textstyle{n\over 2}(1+\lambda
n^{-1/3})) is planar}.$$
  In particular, we obtain $p(0) \approx 0.99780$.
  We extend these results to classes of graphs closed under taking minors. As
an example, we show that the probability of $G(n,\textstyle{n\over 2})$ being
series-parallel converges to 0.98003.
  For the sake of completeness and exposition we reprove in a concise way
several basic properties we need of a random graph near the critical point.
