  The famous Posa conjecture states that every graph of minimum degree at least
2n/3 contains the square of a Hamilton cycle. This has been proved for large n
by Koml\'os, Sark\"ozy and Szemer\'edi. Here we prove that if p >
n^{-1/2+\eps}, then asymptotically almost surely, the binomial random graph
G_{n,p} contains the square of a Hamilton cycle. This provides an `approximate
threshold' for the property in the sense that the result fails to hold if p<
n^{-1/2}.
