The asymptotic capacity at low input powers of an average-power limited or an average- and peak-power limited discrete-time Poisson channel is considered. For a Poisson channel whose dark current is zero or decays to zero linearly with its average input power E, capacity scales like E log 1/E for small E. For a Poisson channel whose dark current is a nonzero constant, capacity scales, to within a constant, like E log log 1/E for small E.