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In this first paper of a two-part series, a single-user single-input single-output (SISO) shot-noise-limited Poisson channel is considered over which an information signal is transmitted by modulating the intensity of an optical beam, and individual photon arrivals are counted at the photodetector receiver. The transmitted signal, which is chosen to satisfy peak and average constraints, undergoes multiplicative fading, which occurs over coherence time intervals of fixed duration. The fade coefficient (channel state) remains constant in each coherence interval, and varies across successive such intervals in an independent and identically distributed (i.i.d.) fashion. A single-letter characterization of the capacity of this channel is obtained when the receiver is provided with perfect channel state information (CSI) while the transmitter CSI can be imperfect. The asymptotic behavior of channel capacity in the low and high peak-signal-to-shot-noise ratio (SNR) regimes is studied.