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.