Talking face generation has been extensively investigated owing to its wide applicability. The two primary frameworks used for talking face generation comprise a text-driven framework, which generates synchronized speech and talking faces from text, and a speech-driven framework, which generates talking faces from speech. To integrate these frameworks, this paper proposes a unified facial landmark generator (UniFLG). The proposed system exploits end-to-end text-to-speech not only for synthesizing speech but also for extracting a series of latent representations that are common to text and speech, and feeds it to a landmark decoder to generate facial landmarks. We demonstrate that our system achieves higher naturalness in both speech synthesis and facial landmark generation compared to the state-of-the-art text-driven method. We further demonstrate that our system can generate facial landmarks from speech of speakers without facial video data or even speech data.