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Oral exams are compelling because (1) students can be asked to provide their reasoning for an answer and (2) their interactive nature permits follow-up questions to resolve ambiguous answers. But oral exams scale poorly. In this paper, we present scalable conversational assessment, an LLM-based system that conducts interactive text-based conversations with students to assess their ability to make and support an argument. We use a two-agent architecture to separate concerns and reduce the likelihood that answers are leaked to the student: the interviewer interacts directly with the student, but only receives guidance about the next question to ask the student; the hidden evaluator knows the goals of the whole activity, tracks the student's progress using a rubric, and guides the interviewer on when to follow-up on the current question or proceed to the next question. The system was first tested using a collection of simulated students with LLM personas of a particular ability level or adversarial tactic (e.g., answer-extractors, prompt-injectors, confident fabricators). A novel element of this evaluation was that it required the use of a fictional assessment topic to prevent personas that shouldn't have information from using knowledge present in the LLM's pre-training. The system was also deployed in a small, elective AI literacy course in both formative and summative contexts. Through manual and post-facto AI review of the transcripts we identified opportunities to improve the follow-up question generation and ensure that grading only credited ideas that the student (and not the interviewer) contributed.