Phase noise is a critical issue for the sensitivity of the interferometric gravitational-wave detector Laser Interferometer Space Antenna that requires laser sources with 1 W of output power. A low-power seed laser with a subsequent fiber amplifier is in discussion, and we report on spectrally resolved phase-noise measurements of an ytterbium-doped fiber amplifier emitting 1 W of output power from 10 mW of seed power. Phase-noise measurements for Fourier frequencies from 10{\^}-4 Hz to 100 kHz show 60 rad/Hz{\^}0.5 at 10{\^}-4 Hz and 0.2 mrad/Hz{\^}0.5 at 1 kHz. This measured excess phase noise is orders of magnitude below the free-running phase noise of nonplanar ring oscillators.