One of the central objects of interest in additive combinatorics is the sumset $A+B= {a+b:a \in A, b \in B}$ of two sets $A,B$ of integers. Our main theorem, which improves results of Green and Morris, and of Mazur, implies that the following holds for every fixed $\lambda > 2$ and every $k>(\log n)^4$: if $\omega$ goes to infinity as $n$ goes to infinity (arbitrarily slowly), then almost all sets $A \subseteq [n]$ with $|A| = k$ and $ |A + A| < \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega$. This is joint work with Marcelo Campos, Mauricio Collares, Rob Morris and Victor Souza.