Fringe projection profilometry (FPP) is a widely used non-contact 3D measurement method. Though maturing in the last decade, it remains a significant challenge when facing the phase unwrapping of measured object surfaces in a single-shot measurement setting. With the rapid development of deep learning techniques, the adoption of a data-driven approach is gaining popularity in the field of optical metrology. This study proposes a new absolute phase recovery method based on the devised single-stage deep learning network. The aim is to ensure high quality absolute phase recovery from a single-shot fringe projection measurement. Unlike most existing approaches, where the numerators and denominators of the wrapped phases and the fringe orders are predicted in various stages, the proposed method acquires the wrapped phases and the corresponding fringe orders within a single network, i.e. it can predict both wrapped phases and the corresponding fringe orders directly and simultaneously from the single fringe pattern projected in the single-shot mode based on a unified Y-shaped network. Experiments on benchmark datasets and models have demonstrated the effectiveness and efficiency of the technique, especially in terms of high-quality recovery of absolute phase information by using the lightweight single-stage network, and enabling the FPP-based phase 3D measurements in an online manner.