Suppose $A$ is a set of non-negative integers with upper Banach density $\alpha$ (see definition below) and the upper Banach density of $A+A$ is less than $2\alpha$. We characterize the structure of $A+A$ by showing the following: There is a positive integer $g$ and a set $W$, which is the union of $\lceil 2\alpha g-1\rceil$ arithmetic sequences [We call a set of the form $a+d\mathbb{N}$ an arithmetic sequence of difference $d$ and call a set of the form $\{a,a+d,a+2d,...,a+kd\}$ an arithmetic progression of difference $d$. So an arithmetic progression is finite and an arithmetic sequence...