We consider the one-dimensional nearest neighbors symmetric simple exclusion process starting with the equilibrium product distribution with density $\rho$. We study $T_N$, the first time for which the interval $\{1,\dots,N\}$ is totally occupied. We show that there exist $0 < \a'\le \a_N\le \a''< \infty$ such that $\a_N\rho^NT_N$ converges to an exponential random variable of mean $1$. More precisely, we get the following uniform sharp bound: $\sup_{t\ge 0}\vert P\{\a_N\rho^N T_N > t\} - e^{-t} \vert \le A\rho^{A'N}$ where $A$ and $A'$ are positive constants independent of $N$.