In many wireless communication systems, radios are subject to duty cycle constraint, that is, a radio only actively transmits signals over a fraction of the time. For example, it is desirable to have a small duty cycle in some low power systems; a half-duplex radio cannot keep transmitting if it wishes to receive useful signals; and a cognitive radio needs to listen and detect primary users frequently. This work studies the capacity of scalar discrete-time Gaussian channels subject to duty cycle constraint as well as average transmit power constraint. The duty cycle constraint can be regarded as a requirement on the minimum fraction of nontransmission or zero symbols in each codeword. A unique discrete input distribution is shown to achieve the channel capacity. In many situations, numerical results demonstrate that using the optimal input can improve the capacity by a large margin compared to using Gaussian signaling over a deterministic transmission schedule, which is capacity-achieving in the absence of the duty cycle constraint. This is in part because the positions of the nontransmission symbol in a codeword can convey information. The results suggest that, under the duty cycle constraint, departing from the usual paradigm of intermittent packet transmissions may yield substantial gain.