The large-inputs asymptotic capacity of a peak and average power limited discrete-time Poisson channel is derived using a new firm (non-asymptotic) lower bound and an asymptotic upper bound. The upper bound is based on the dual expression for channel capacity and the recently introduced notion of "capacity-achieving input distributions that escape to infinity." The lower bound is based on a lemma that lower bounds the entropy of a conditionally Poisson random variable in terms of the differential entropy of the conditional mean.