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In this paper, we consider a cognitive radio (CR) network where a secondary (cognitive) user shares the spectrum for transmission with a primary (non-cognitive) user over block-fading (BF) channels. It is assumed that the primary user has a constant-rate, constant-power transmission, while the secondary user is able to adapt transmit power and rate allocation over different fading states based on the channel state information (CSI) of the CR network. We study a new type of constraint imposed over the secondary transmission to protect the primary user by limiting the maximum transmission outage probability of the primary user to be below a desired target. We derive the optimal power allocation strategies for the secondary user to maximize its ergodic/outage capacity, under the average/peak transmit power constraint along with the proposed primary user outage probability constraint. It is shown by simulations that the derived new power allocation strategies can achieve substantial capacity gains for the secondary user over the conventional methods based on the interference temperature (IT) constraint to protect the primary transmission, with the same resultant primary user outage probability.