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We report a 1.04 GHz high-power Kerr-lens mode-locked Yb:CaYAlO4 laser pumped by a single-mode fiber laser at 976 nm. Based on a bow-tie cavity, stable unidirectional mode-locked operation is obtained with an output coupler of 1.6%. The oscillator delivers pulses with an average power of 1.46 W and with the pulse duration of 99 fs, which, to the best of our knowledge, is the first gigahertz-level Kerr-lens mode-locked laser based on the Yb:CaYAlO4 gain medium. We believe that the watt-level solid-state femtosecond laser at GHz would be an excellent source for developing time-resolved broadband dual-comb spectroscopy.