We have developed a frequency-domain terahertz spectrometer based on homebuilt 1 μm band external cavity diode lasers, for high resolution spectroscopy. Our spectrometer is digitally controlled to a resolution of 10 MHz, and uses InGaAs/GaAs photoconductive antennas. We have obtained a spectrum in the range 0.02 THz to 2.5 THz, which exceeds the conventional temperature tuning range of a distributed feedback diode laser. We achieved a signal-to-noise ratio of up to 80 dB at around 0.05 THz, and 20 dB at around 2.0 THz. We observed water vapor spectra in the atmosphere with a frequency step of 0.6 GHz in the region between 1.0 THz and 2.0 THz. We have demonstrated that our 1 μm-band frequency-domain terahertz spectrometer is competitive when compared with existing 800 nm- and 1.5 μm-band systems.