In this paper, we present a Deep Reinforcement Learning (DRL) based real-time smooth UAV motion planning method for solving catastrophic flight trajectory oscillation issues. By formalizing the original problem as a linear mixture of dual-objective optimization, a novel Deep smOoth Motion plAnning (DOMA) algorithm is proposed, which adopts an alternative layer-by-layer gradient descending optimization approach with the major gradient and the DOMA gradient applied separately. Afterward, the mix weight coefficient between the two objectives is also optimized adaptively. Experimental result reveals that the proposed DOMA algorithm outperforms baseline DRL-based UAV motion planning algorithms in terms of both learning efficiency and flight motion smoothness. Furthermore, the UAV safety issue induced by trajectory oscillation is also addressed.