Spoken medical dialogue is a valuable source of information for patients and caregivers. This work presents a first step towards automatic analysis and summarization of spoken medical dialogue. We first abstract a dialogue into a sequence of semantic categories using linguistic and contextual features integrated in a supervised machine-learning framework. Our model has a classification accuracy of 73%, compared to 33% achieved by a majority baseline (p<0.01). We then describe and implement a summarizer that utilizes this automatically induced structure. Our evaluation results indicate that automatically generated summaries exhibit high resemblance to summaries written by humans. In addition, task-based evaluation shows that physicians can reasonably answer questions related to patient care by looking at the automatically generated summaries alone, in contrast to the physicians' performance when they were given summaries from a naïve summarizer (p<0.05). This work demonstrates the feasibility of automatically structuring and summarizing spoken medical dialogue.