A finite-time adaptive distributed control algorithm for second-order, double integrator-type agents is suggested to achieve consensus with a second order controlled leader. For this, a recently suggested finite-time adaptive estimator is designed for each agent using only local agent information. This adaptive estimator is driven via direct parameter estimation error information and a nonlinear formulation of the consensus error, which guarantees finite-time consensus. The parameter error information, computed through auxiliary filters for each agent, ensures convergence via a sliding-mode type unit vector in the estimation algorithm. This algorithm is suitably combined with a finite-time consensus control algorithm, enhancing recently suggested adaptive consensus algorithms in terms of robustness, parameter estimation and convergence speed. Suitable comparative simulation examples provide practically relevant evidence.