  A family of sets is called union-closed if whenever $A$ and $B$ are sets of
the family, so is $A\cup B$. The long-standing union-closed conjecture states
that if a family of subsets of $[n]$ is union-closed, some element appears in
at least half the sets of the family. A natural weakening is that the
union-closed conjecture holds for large families; that is, families consisting
of at least $p_02^n$ sets for some constant $p_0$. The first result in this
direction appears in a recent paper of Balla, Bollob\'as and Eccles
\cite{BaBoEc}, who showed that union-closed families of at least
$\frac{2}{3}2^n$ sets satisfy the conjecture --- they proved this by
determining the minimum possible average size of a set in a union-closed family
of given size. However, the methods used in that paper cannot prove a better
constant than $\frac{2}{3}$. Here, we provide a stability result for the main
theorem of \cite{BaBoEc}, and as a consequence we prove the union-closed
conjecture for families of at least $(\frac{2}{3}-c)2^n$ sets, for a positive
constant $c$.
