We present a SESAM modelocked Yb:YAG solid-state laser providing low-noise narrowband pulses with a pulse duration of 606 fs at a 1.09-GHz repetition rate, delivering up to 2.5 W of average output power. This laser provides access to a new parameter space that could previously not be reached by solid-state lasers and, to the best of our knowledge, is the first modelocked solid-state Yb:YAG laser in the gigahertz regime. This is achieved by introducing a single additional intracavity element, specifically a nonlinear birefringent YVO4 crystal, for soliton formation, polarization selection, and cavity intensity clamping. The isotropic pump absorption in Yb:YAG allows for stable and low-noise operation with multimode fiber pumping. This laser is ideally suited as a seed source for many commercial high-power Yb-doped amplification systems operating at a center wavelength around 1.03 µm. The laser exhibits a high power per comb line of 5.0 mW which also makes it interesting for applications in frequency comb spectroscopy, especially if it is used to pump an optical parametric oscillator. We measure a relative intensity noise (RIN) of 0.03%, integrated from 1 Hz to 10 MHz. Furthermore, we show that the laser timing jitter for noise frequencies >2 kHz is fully explained by a power-dependent shift in the center wavelength of 0.38 nm/W due to the quasi-three-level laser gain material. The narrow gain bandwidth of Yb:YAG reduces this contribution to noise in comparison to other SESAM modelocked Yb-doped lasers.