Online peer-to-peer support platforms enable conversations between millions of people who seek and provide mental health support. If successful, web-based mental health conversations could improve access to treatment and reduce the global disease burden. Psychologists have repeatedly demonstrated that empathy, the ability to understand and feel the emotions and experiences of others, is a key component leading to positive outcomes in supportive conversations. However, recent studies have shown that highly empathic conversations are rare in online mental health platforms. In this paper, we work towards improving empathy in online mental health support conversations. We introduce a new task of Empathic Rewriting which aims to transform low-empathy conversational posts to higher empathy. Learning such transformations is challenging and requires a deep understanding of empathy while maintaining conversation quality through text fluency and specificity to the conversational context. Here we propose Partner, a deep reinforcement learning (RL) agent that learns to make sentence-level edits to posts in order to increase the expressed level of empathy while maintaining conversation quality. Our RL agent leverages a policy network, based on a transformer language model adapted from GPT-2, which performs the dual task of generating candidate empathic sentences and adding those sentences at appropriate positions. During training, we reward transformations that increase empathy in posts while maintaining text fluency, context specificity, and diversity. Through a combination of automatic and human evaluations we demonstrate that our approach successfully generates more empathic, diverse, and context-specific responses and outperforms current natural language processing methods from related tasks such as style transfer and sequence-to- sequence generation. This work has direct implications for facilitating empathetic conversations on web-based platforms.