We demonstrate GHz-repetition rate mode-locked operation of a femtosecond-laser-inscribed Yb:YAG channel waveguide laser using single-walled carbon nanotube saturable absorber mirror (SWCNT-SAM). A 6.3-mm-long, type II Yb:YAG waveguide laser with an extended cavity configuration delivers mode-locked picosecond (ps) pulses at GHz repetition rates. The dispersion of the laser cavity is compensated by the combination of a multi-functional output coupler and the Gires-Tournois interferometer (GTI) effect arising from an air-gap between the facet of the waveguide and the output coupler. The incident beam fluence on the SWNCNT-SAM is controlled by adjusting two intracavity lenses to avoid optical damage on the polymer nanocomposite matrix containing the SWCNTs. The average output power of our mode-locked waveguide laser is 322 mW at a pump power of 3.2 W. Nearly Fourier-limited, stable 2-ps-short pulses are generated at a repetition rate of 2.08 GHz.