Ransomware operations have evolved from relatively unsophisticated threat actors into highly coordinated cybercrime syndicates that regularly extort millions of dollars in a single attack. Despite dominating headlines and crippling businesses across the globe, there is relatively little in-depth research into the modern structure and economics of ransomware operations. In this paper, we leverage leaked chat messages to provide an in-depth empirical analysis of Conti, one of the largest ransomware groups. By analyzing these chat messages, we construct a picture of Conti's operations as a highly-profitable business, from profit structures to employee recruitment and roles. We present novel methodologies to trace ransom payments, identifying over $80 million in likely ransom payments to Conti and its predecessor -- over five times as much as in previous public datasets. As part of our work, we publish a dataset of 666 labeled Bitcoin addresses related to Conti and an additional 75 Bitcoin addresses of likely ransom payments. Future work can leverage this case study to more effectively trace -- and ultimately counteract -- ransomware activity.