The paper describes the determination of the additive white Gaussian noise channel information transmission capacity when the channel input and output are limited by certain constraints. The channel input constraints are those of signal amplitude, or signal amplitude and average power. The input signal amplitude and average power constraints are defined by restricting the channel input to values within the finite interval [—A, +A] and also to have average power equal to some specified value. The channel output constraint is that of signal clipping due to quantisation applied at the receiver. The input signal amplitude/output signal clipping constrained capacity, and the input signal amplitude and average power/output signal clipping constrained capacity are determined separately. It is found that there are unique, optimum and discrete input signal amplitude distributions, taking a finite number of values, and optimum output signal clippings that achieve these capacity values. The optimum input distribution values are also used to determine the optimum amplitude probability density function at the channel output.