We demonstrate a rapid scanning high-resolution THz spectrometer capable of acquiring THz field transients with 1 ns duration without mechanical delay line. The THz spectrometer is based on two 1-GHz Ti:sapphire femtosecond lasers which are linked with a fixed repetition rate difference in order to perform high-speed asynchronous optical sampling. One laser drives a high-efficiency large-area GaAs based THz emitter, the other laser is used for electro-optic detection of the emitted THz-field. At a scan rate of 9 kHz a time resolution of 230 fs is accomplished. High-resolution spectra from 50 GHz up to 3 THz are obtained and water absorption lines with a width of 11 GHz are observed. The use of femtosecond lasers with 1 GHz repetition rate is essential to obtain rapid scanning and high time-resolution at the same time.