We demonstrate generation of continuous-wave terahertz radiation and its application to molecular spectroscopy. The radiation source is based on the photomixing of two diode laser beams in a low-temperature-grown GaAs photoconductive antenna, which offered output power of several tens of nanowatts at frequencies up to 2 THz with its long term frequency stability being about 5 MHz. The pure rotational spectra of CH3CN and isotopomers of CO were measured with this source, and 1% of absorption was clearly detected with a simple amplitude modulation technique. This indicates that the present system has a potential capability for high-resolution and high-sensitivity molecular spectroscopy.