This paper proposes an immersion and invariance theory-based prescribed-time adaptive control method for a series elastic actuator system with parametric uncertainties. A newly designed time-scale transformation function is employed to overcome the singularity problem caused by traditional time-scale transformations near the prescribed time. Then, the parameter estimation structure of the immersion and invariance method is introduced, and a prescribed-time convergent adaptive law is developed to guarantee the prescribed-time convergence of the parameter estimation error. Furthermore, a prescribed-time sliding mode surface is designed to ensure that the tracking error of the system converges to a given small neighborhood within the prescribed time. Besides, the stability of the closed-loop system is validated via the Lyapunov method. Simulation results verify the effectiveness of the proposed method.