This paper addresses the three-dimensional trajectory tracking problem of underactuated autonomous underwater vehicles (AUVs) operating in the presence of external disturbances and unmodeled dynamics by proposing a predefined-time adaptive control scheme. Firstly, the underactuated AUV system was decoupled into drive and non-drive subsystems to facilitate the design of a controller that does not rely on specific model parameters. Radial basis function neural networks (RBFNNs) were employed to estimate the external disturbances. To enhance tracking performance, a predefined-time adaptive control law was designed to ensure that tracking errors converged to a small neighborhood around the origin within the predefined time. The adaptive control law compensated for the unmodeled components. Finally, we used theoretical proofs and simulations to show that our method is effective and superior.