An all-fiber, narrow-linewidth, high power Yb-doped silica fiber laser at 1179 nm has been demonstrated. More than 12 W output power has been obtained, corresponding to a slope efficiency of 43% with respect to launched pump power, by core-pumping at 1090 nm. In order to increase the pump absorption, the Yb-doped fiber was heated up to 125 degrees C. At the maximum output power, the suppression of amplified spontaneous emission was more than 50 dB. Furthermore, theoretical work confirms that the proposed laser architecture can be easily scaled to higher power.