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 $|A|+|B|-1$.
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