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The issue of physical layer security in single-antenna wiretap channels with slow fading is addressed in this letter. With the assumption of perfect channel state information (CSI) about the main channel and statistical CSI about the eavesdropper's channel, we propose a new secure transmission scheme under the constraints of transmit power and secrecy outage probability (SOP). This scheme fixes the communication rate and adaptively adjusts the secrecy rate and the transmit power during transmission, which is different from the conventional adaptive and non-adaptive schemes in the literature. By fixing the communication rate, the proposed scheme fixes the length of codebook and thus simplifies the codebook design. Simulation results show that the proposed scheme performs well when the SOP constraint is strict.