In this paper, we consider a secondary link sharing the spectrum with a primary link in a fading cognitive radio (CR) network. Instead of applying the conventional interference power constraint at the primary user (PU) receiver for the secondary user (SU) to protect the primary transmission, we propose a new constraint on the maximum tolerable outage probability for the PU due to the SU transmission. Under the assumption that perfect instantaneous channel state information (CSI) on the SU channel, the channel from the SU transmitter to PU receiver, and the PU channel is available at the SU transmitter, we derive the optimal power allocation strategies to achieve the ergodic capacity of the SU fading channel. It is shown by simulations that the proposed power allocation strategies can achieve substantial capacity gain for the SU over that based on the conventional interference power constraint, for the same PU outage probability loss. 1