A large number of musculoskeletal and neurological disorders can affect the upper limb limiting the subject's ability to perform activities of daily living. In recent years, rehabilitation therapies based on robotics have been proposed as complement to the work of therapists. This paper introduces a prototype of exoskeleton for the evaluation and rehabilitation therapy of the elbow joint in flexion extension and pronation-supination. The main novelty is the use of bioinspired actuators based on shape memory alloys (for the first time) in an upper limb rehabilitation exoskeleton. Because of this, the device presents a light weight, less than 1 kg, and noiseless operation, both characteristics are very beneficial for rehabilitation therapies. In addition, the prototype has been designed with low-cost electronics and materials, and the result is a wearable, comfortable, and cheap rehabilitation exoskeleton for the elbow joint. The exoskeleton can generate the joint torque (active mode) or it can be used as a passive tool. (The patient performs therapy by itself, carrying the device while it collects relevant movement data for evaluation.) The simulations and experimental tests validate the solution in the first phases of rehabilitation therapies when slow and repetitive movements are required.