Matter in its liquid state is convenient for processing and controlling chemical reactions, owing to its fluidity. Recently much evidence has been accumulated for the existence of a liquid-liquid transition (LLT) in single-component liquids. Here, we report that we can control, by the LLT of a molecular liquid, triphenyl phosphite (TPP), the fluidity and miscibility of its mixture with another molecular liquid. For a mixture of TPP with toluene or aniline, we find that both liquid I and II mix well and liquid II remains in a 'liquid' state, in contrast to pure TPP, where liquid II is a non-ergodic amorphous state. This is the first example of a 'true' LLT in a molecular liquid. Furthermore, we find demixing induced by the LLT for a mixture of TPP with diethyl ether or ethanol. These findings will open a new phase of research towards various applications of the LLT.