Robots that can talk with humans play increasingly important roles in society. However, current conversation robots remain unskilled at eliciting empathic feelings in humans. To address this problem, we used a robot that speaks in a voice synchronized with human vocal prosody. We conducted an experiment in which human participants held positive conversations with the robot by reading scenarios under conditions with and without vocal synchronization. We assessed seven subjective responses related to affective empathy (e.g., emotional connection) and measured the physiological emotional responses using facial electromyography from the corrugator supercilii and zygomatic major muscles as well as the skin conductance level. The subjective ratings consistently revealed heightened empathic responses to the robot in the synchronization condition compared with that under the de-synchronizing condition. The physiological signals showed that more positive and stronger emotional arousal responses to the robot with synchronization. These findings suggest that robots that are able to vocally synchronize with humans can elicit empathic emotional responses.