We prove that if T is a tree on n vertices with maximum degree D and the edge probability p(n) satisfies:
np c maxD*log n,nε
for some positive ε0, then with high probability (w.h.p.) the random graph G(n,p) contains a copy of T. In particular, G(n,n-1+ε) w.h.p. contains a copy of any given bounded degree tree T on n vertices. The obtained bound on the edge probability is shown to be essentially tight for D=nΘ(1).