Entrainment, the responsive communication between interacting individuals, is a crucial process in building a strong relationship between a mental health therapist and their client, leading to positive therapeutic outcomes. However, so far entrainment has not been investigated as a measure of efficacy of large language models (LLMs) delivering mental health therapy. In this work, we evaluate the linguistic entrainment of an LLM (ChatGPT 3.5-turbo) in a mental health dialog setting. We first validate computational measures of linguistic entrainment with two measures of the quality of client self-disclosures: intimacy and engagement (p < 0.05). We then compare the linguistic entrainment of the LLM to trained therapists and non-expert online peer supporters in a cognitive behavioral therapy (CBT) setting. We show that the LLM is outperformed by humans with respect to linguistic entrainment (p < 0.001). These results support the need to be cautious in using LLMs out-of-the-box for mental health applications.