In this paper, we propose an adaptive control scheme for achieving finite-time regulation in the simplest nonlinear mechanical system: a one degree of freedom Euler-Lagrange system. Our control approach consists of two components: a Proportional-Derivative (PD) nonlinear feedback and a finite-time estimation algorithm, which is based on a Dynamic Regressor Extension and Mixing (DREM) estimator. We emphasize that finite-time regulation of Euler-Lagrange systems using adaptive control has not been addressed in the literature. Although the control system under analysis is basic, it reveals important aspects of the finitetime convergence of the closed-loop system. The performance of the proposed controller is demonstrated through simulations.