For a Poisson distribution with parameter a and median M, it is known that the best bounds for M-a are given by –log(2) and 1/3. Asymptotically, it is even shown that those bounds become -2/3 and 1/3. Aiming to obtain an asymptotic distribution for the empirical median of a Poisson sample, we studied the median of a random variable obtained by adding a Poisson distribution with parameter a and a uniform distribution defined on [0,1]. Surprisingly, we were able to show that the median of such a variable is close to a+1/3, and is even closer as a gets bigger. Afterwards, by a simulation study, we have compared the estimator of a from this procedure with other existing robust estimators of the parameter a.