This paper presents the design, control, and experimental validation of a novel fully–actuated aerial robot for physically interactive tasks, named Tilt-Hex. We show how the Tilt-Hex, a tilted-propeller hexarotor, is able to control the full pose (position and orientation independently) using a geometric control, and to exert a full-wrench (force and torque independently) with a rigidly attached end-effector using an admittance control paradigm. An outer loop control governs the desired admittance behavior and an inner loop based on geometric control ensures pose tracking. The interaction forces are estimated by a momentum based observer. Control and observation are made possible by a precise control and measurement of the speed of each propeller. An extensive experimental campaign shows that the Tilt-Hex is able to outperform the classical underactuated multi-rotors in terms of stability, accuracy and dexterity and represents one of the best choice at date for tasks requiring aerial physical interaction.