  Let $A, B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min
B=0$, and let $$\delta(A,B)={\begin{array}{ll}
  1 & \hbox{if} A\subseteq B,
  0 & \hbox{otherwise.}
  If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq
|A|+2|B|-3-\delta(A,B), then we show $A+B$ contains an arithmetic progression
with difference 1 and length $|A|+|B|-1$.
  As a corollary, if \eqref{one} holds, $\max(B)\leq \max(A)$ and either
$\gcd(A)=1$ or else $\gcd(A+B)=1$ and $|A+B|\leq 2|A|+|B|-3$, then $A+B$
contains an arithmetic progression with difference 1 and length $|A|+|B|-1$.
