The effects of radial advection and thermal diffusion were considered in investigating the linear stability of an optically thin, two-temperature accretion disc. If the disc has very little advection, we prove that the thermal instability exists when the disc is geometrically thin. However, it disappears in a geometrically slim disc if the thermal diffusion is considered. Moreover, if the disc is advection-dominated, the thermal instability does not exist. We also found that the instabilities of inertialacoustic modes exist only in a geometrically thin disc or an advection-dominated disc with a low Mach number, whereas the Lightman & Eardley viscous instability always disappears in a two-temperature disc. A simple comparison also showed that an optically thin, bremsstrahlung cooling-dominated disc is generally more thermally unstable than a two-temperature disc if it is not advection-dominated.