Talking face generation is the task of synthesizing a natural face synchronous to driving audio. Although much progress has been made in terms of visual quality, lip synchronization, and facial motion of the talking face, current works still struggle to overcome issues of crude and asynchronous lip movement, which can result in puppetry-like animation. We identify that the prior works commonly correlate lip movement with audio at the phone level. However, due to co-articulation, where an isolated phone is influenced by the preceding or following phones, the articulation of a phone varies upon the phonetic context. Therefore, modeling lip motion with the phonetic context can generate more spatio-temporally aligned and stable lip movement. In this respect, we investigate the phonetic context in lip motion for authentic talking face generation. We propose a Context-Aware Lip-Sync framework (CALS), which leverages phonetic context to generate more spatio-temporally aligned and stable lip movement. The CALS comprises an Audio-to-Lip module and a Lip-to-Face module. The former explicitly maps each phone to a contextualized lip motion unit, which guides the latter in synthesizing a target identity with context-aware lip motion. In addition, we introduce a discriminative sync critic that enforces accurate lip displacements within the phonetic context through audio-visual sync loss and visual discriminative sync loss. From extensive experiments on LRW, LRS2, and HDTF datasets, we demonstrate that the proposed CALS effectively enhances spatio-temporal alignment, greatly improving upon the state-of-the-art on visual quality, lip-sync quality, and realness. Finally, we show the authenticity of the generated video through a lip readability test and achieve 97.7% of relative word prediction accuracy to real videos.