This paper addresses the problem of designing and experimentally validating a controller for steering a quadrotor vehicle along a trajectory, while rejecting wind disturbances. The proposed solution consists of a nonlinear adaptive state feedback controller for thrust and torque actuation that asymptotically stabilizes the closed-loop system in the presence of constant force disturbances, used to model the wind action, and ensures that the actuation does not grow unbounded as a function of the position errors. A prototyping and testing architecture, developed to streamline the implementation and the tuning of the controller, is also described. Experimental results are presented to demonstrate the performance and robustness of the proposed controller.