We prove that there is a small but fixed positive integer $ϵ$ such that for every prime $p$ larger than a fixed integer, every subset $S$ of the integers modulo $p$ which satisfies $\left|2S\right|\le (2+ϵ)\left|S\right|$ and $2\left(\right|2S\left|\right)-2\left|S\right|+3\le p$ is contained in an arithmetic progression of length $\left|2S\right|-\left|S\right|+1$. This is the first result of this nature which places no unnecessary restrictions on the size of $S$.