The first time that the $N$ sites to the right of the origin get empty in a one dimensional zero range process is shown to converge, as $N\to\infty$ to the exponential distribution, when divided by its mean. The initial distribution of the process is assumed to be one of the extremal invariant measures $\nur$, $\rho\in (0,1)$ with density $\rho/(1-\rho)$. The proof is based on the classical Burke's theorem. \vskip 2truemm \noindent {\sl Keywords.} Zero range process, occurrence time of a rare event, large deviations. \vskip 1truemm \noindent {\sl AMS 1991 classification numbers.} 60K35, 82C22, 60F10.
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