  The \emph{spanning tree packing number} of a graph $G$ is the maximum number
of edge-disjoint spanning trees contained in $G$. Let $k\geq 1$ be a fixed
integer. Palmer and Spencer proved that in almost every random graph process,
the hitting time for having $k$ edge-disjoint spanning trees equals the hitting
time for having minimum degree $k$. In this paper, we prove that for any $p$
such that $(\log n+\omega(1))/n\leq p\leq (1.1\log n)/n$, almost surely the
random graph $G(n,p)$ satisfies that the spanning tree packing number is equal
to the minimum degree. Note that this bound for $p$ will allow the minimum
degree to be a function of $n$, and in this sense we improve the result of
Palmer and Spencer. Moreover, we also obtain that for any $p$ such that $p\geq
(51\log n)/n$, almost surely the random graph $G(n,p)$ satisfies that the
spanning tree packing number is less than the minimum degree.
