This paper introduces UniGen, a novel approach to generating new traffic scenarios for evaluating and improving autonomous driving software through simulation. Our approach models all driving scenario elements in a unified model: the position of new agents, their initial state, and their future motion trajectories. By predicting the distributions of all these variables from a shared global scenario embedding, we ensure that the final generated scenario is fully conditioned on all available context in the existing scene. Our unified modeling approach, combined with autoregressive agent injection, conditions the placement and motion trajectory of every new agent on all existing agents and their trajectories, leading to realistic scenarios with low collision rates. Our experimental results show that UniGen outperforms prior state of the art on the Waymo Open Motion Dataset.