In this paper, we develop a novel adaptive motion imitation algorithm (AMI) for robotic systems. Although AMI can be used in a variety of human-robot interaction scenarios, we are particularly interested in robotic rehabilitation where the robot plays the role of demonstrating and practicing challenging motion physiotherapy. During therapy, the robot first demonstrates a reference trajectory to the patient that needs to be repeated during practice and then adapts its motion to a cyclic speed and amplitude based on the patient's abilities. Using this algorithm, the robotic system learns an upper-body motion of the human user and performs a unique, similar, and easier motion based on the learned trajectory from the user. Adaptation in the AMI is based on deep reinforcement learning with deep deterministic policy gradient implemented in the Robot Operating System (ROS) environment. Experimental data collected from 11 users during upper body human-robot imitation sessions with social robot Zeno was used to show that the algorithm can learn reference elbow joint trajectories of the user in an off-line manner after just a few cycles. Finally, we also implemented the algorithm online using the Baxter robot to demonstrate its learning and playback performance.