  Merging together a result of Nathanson from the early 70s and a recent result
of Granville and Walker, we show that for any finite set $A$ of integers with
$\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail",
such that if $m\ge\max(A)-|A|+2$, then the $m$-fold sumset $mA$ consists of the
union of these sets and a long block of consecutive integers separating them.
We give sharp estimates for the length of the block, and investigate the
corresponding stability problem classifying those sets $A$ for which the bound
$\max(A)-|A|+2$ cannot be substantially improved.
