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A proper gamma approximation to a Poisson Jeffreys prior uses shape $1/2$ and a small positive rate $\epsilon$:Its density is proportional to ${\lambda }_i^{-1/2}{e}^{-\epsilon {\lambda }_i}$, so where $\epsilon {\lambda }_i$ is small its kernel approximates the Jeffreys prior. After observing the data, Poisson-gamma conjugacy gives $\mathrm{Gamma}(y_i+1/2,E_i+\epsilon )$, which converges to the preceding posterior distribution as $\epsilon \downarrow 0$. This is an approximation of an improper kernel on the relevant parameter range, not a weak limit to a normalized Jeffreys probability distribution. The shape must be $1/2$; a small-shape gamma prior would approximate a different power of ${\lambda }_i$.