A single-mode frequency-stabilized broadly-tunable narrow-linewidth external cavity diode laser (ECL) is demonstrated. A broadly coarse wavelength tuning range of 135 nm for the spectral range between 1250 and 1385 nm is achieved by rotating the diffraction grating forming a Littrow-type external-cavity configuration. The laser is locked to a Fabry-Perot interferometer with a free spectral range of 1.5 GHz. The full width at half maximum is narrowed from 450 to 2.9 kHz by the laser frequency locking. Our system is suitable for applications or experiments requiring a frequency-stabilized ECL with narrow linewidth, such as coherent population trapping, precision spectroscopy, optical clocks, atom/ion cooling, Bose-Einstein condensates, atom interferometry, and quantum memory.