This paper presents the mechanical design of an ultra-lightweight, biomimetically actuated anthropomorphic hand for prosthetic purposes. The proposed design is based on an underactuated configuration of 16 joints and 7 active degrees of freedom. Shape Memory Alloy wires are used as motive elements of a specially designed actuation system installed within the envelope of the hand and forearm; due to their inherent contraction ability when heated, these innovative micro-actuators produce linear motion which is imparted to the fingers via a tendon transmission system. The overall design is completed with the integration of the necessary locking mechanisms for power saving. An experimental prototype of the suggested prosthesis has been fabricated using rapid prototyping techniques and it will be used as an evaluation platform for further research towards the development of a truly multifunctional, silent and cosmetically appealing hand for upper limb amputees.