Based on an optimized underactuated finger using rolling contact joints, a novel prosthetic hand is proposed in this paper to increase the adaptability to objects of highly underactuated grippers. The hand has 17 degrees-of-freedom and only one degree of actuation, yielding a robust and simple control. A manually operated thumb, able to reach three stable positions corresponding to the main grasping postures is first presented. Its rolling contact joints, introduce geometric twist and tilt, are designed in an iterative process using 3D modeled contacts. In parallel, the parameters of the thumb are optimized to maximize the distribution of contacts and the ability to hold objects. A joint located inside the hand allows the palm to arch, increasing the opposition of the ring finger, and the little finger to the thumb. The floating mechanism used to distribute the single actuation to the different degrees-of-freedom reduces friction while avoiding the unnecessary blocking of components. Finally, a prototype is designed and built to demonstrate the adaptation capabilities of the hand to diverse objects, which is enhanced by a gravity-dependent closing sequence and the deformation of the palm that maximizes the contribution of each finger. Novel possibilities are also presented such as the grasping and flipping of small objects on a hard surface.