Dialogue systems for mental health care aim to provide appropriate support to individuals experiencing mental distress. While extensive research has been conducted to deliver adequate emotional support, existing studies cannot identify individuals who require professional medical intervention and cannot offer suitable guidance. We introduce the Diagnostic Emotional Support Conversation task for an advanced mental health management system. We develop the DESC dataset to assess depression symptoms while maintaining user experience by utilizing task-specific utterance generation prompts and a strict filtering algorithm. Evaluations by professional psychological counselors indicate that DESC has a superior ability to diagnose depression than existing data. Additionally, conversational quality evaluation reveals that DESC maintains fluent, consistent, and coherent dialogues.